Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

Stability of an annular plate of a shape memory alloy

Analytical solutions using various hypotheses are obtained for the problem of buckling of an annular circular plate of a shape memory alloy in the forward or reverse thermoelastic phase transformations under uniform radial compression. It is established that, despite the axial symmetry of the body geometry, loads, and boundary conditions, the minimum critical loads for annular plates correspond to nonaxisymmetric buckling modes, in contrast to continuous plates.     

Winkler地基上变厚度圆板的轴对称弯曲

本文提出了Winkler地基上变厚度圆板轴对称弯曲的传递矩阵算法。首先,根据贝塞尔函数理论获得了等厚度圆板和环板单元在任意荷载作用下轴对称弯曲的解析解,这些解均由通解和特解两部分组成。基于这些解析解,导出了等厚度圆板和环板单元的传递矩阵。然后沿径向将变厚度圆板划分成一个等厚度圆板单元和一系列等厚度环板单元,应用传递矩阵算法原理获得了变厚度圆板的整体传递矩阵。引入圆板的边界条件,给出了该板每条节线上的挠度、径向转角、径向弯矩和径向剪力。最后,讨论了受均布荷载作用的简支线性变厚度圆板的弯曲,将本文数值解与解析解进行比较,证实了本文方法的有效性,并简要地讨论了地基参数对板挠度和径向弯矩的影响。     

Smoke-wire flow visualization of the near-wake region behind a circular disc at low Reynolds numbers

The flow structures in the near field of the unducted wake region behind a circular disc for annular flow at low Reynolds numbers were studied by smoke-wire flow visualization technique. A twisted-dual-wire was employed to perform the time evolving visualization. Three typical characteristic flow modes: Q-tip, open-top toroid, and closed toroid, were identified in the near disc region. For Reynolds number between 130 and 390, the Q-tip flow mode which subject to a periodic up-down oscillatory motion was observed. The open-top toroid mode which experiences the expelling vortex shedding was found for Reynolds number between 390 and 455. The free separation surface turns around and merges to the central axisymmetric axis to form the conventionally observed toroidal recirculation bubble for Reynolds number higher than 455. The closed toroid mode exhibits both expelling and shear-layer vortex sheddings. With the identified flow modes at low Reynolds numbers, the recirculation contours, recirculation length, and the shedding frequency in each mode were measured and discussed.List of symbols B.R. blockage ratio (=D ² /D _a ² ) - D _a outer diameter of annular jet, 30 mm - D diameter of circular disc, 20 mm - f frequency of vortex shedding, Hz - L _r axial length of recirculation zone - R radius of circular disc, 10 mm - u _a average exit velocity of annular jet - ₀ stream function with value of zero - mass density of annular flow - u average axial velocity - r radial coordinate, originated from center of circular disk - r ₀ radial coordinate of the boundary of the recirculation zone - Re _a Reynolds number of annular jet based on the disc diameter - Z axial coordinate, originated from center of circular disk - w _max maximum half-width of the recirculation zone - St Strouhal number (=fD/D _a )     

Frequency equations for the in-plane vibration of orthotropic circular annular plate

Orthotropic circular annular plates have a lot of applications in engineering such as space structures and rotary machines. In this paper, frequency equations for the in-plane vibration of the orthotropic circular annular plate for general boundary conditions were derived. To obtain the frequency equation, first the equation of motion for the circular annular plate in the cylindrical coordinate is derived by using the stress-strain- displacement expressions. Helmholtz decomposition is used to uncouple the equations of motion. The wave equation is obtained by assumption a harmonic solution for the uncoupled equations. Using the separation of the variables leads to the general wave equation solution and the in-plane displacements in the r and θ directions. Finally, boundary conditions are exerted and the natural frequency is derived for general boundary conditions. The obtained results are validated by comparing with the previously reported and those from finite element analysis.     

Rotationally Symmetric Deformations of Partially Loaded Annular Elastic Membranes Without Wrinkling

This paper investigates axisymmetric deformations of curved annular membranes subjected to a partially vanishing vertical surface load and to radial edge loads or displacements. The frame of the membrane model we deal with is the nonlinear small-strain theory. The determination of the principal stresses reduces to the solution of a single nonlinear second order ODE. Analysis becomes explicit on the unloaded membrane part while the loaded part is treated by methods which have been previously developed. In particular, the ranges of those stress and displacement boundary data are determined which admit for wrinkle-free solutions only, i.e. for solutions governed by a nonnegative radial and circumferential stress component. For such a tensile state, a curved membrane flattens out on the unloaded portions. This revised version was published online in August 2006 with corrections to the Cover Date.     

Nonlinear vibrations and instabilities of a stretched hyperelastic annular membrane

The mathematical modeling for the nonlinear vibration analysis of a pre-stretched hyperelastic annular membrane under finite deformations is presented. The membrane is initially fixed along the inner boundary and then subjected to a uniform radial traction along its outer circumference and fixed along the outer boundary. The pre-stretched membrane in then subjected to a transversal harmonic pressure. The membrane material is assumed to be homogeneous, isotropic, and neo-Hookean. First, the solution of the radially stretched membrane is obtained analytically and numerically by the shooting method. The equations of motion of the stretched membrane are then obtained. By analytically and numerically solving the linearized equations of motion, the vibration modes and frequencies of the hyperelastic membrane are obtained, and these normal modes are used, together with the Galerkin method, to obtain reduced order models for the nonlinear dynamic analysis. A parametric analysis of the nonlinear frequency-amplitude relations, resonance curves, bifurcation diagrams and basins of attraction show the influence of the initial stretching ratio and membrane geometry on the type and degree of nonlinearity of the hyperelastic membrane under large amplitude vibrations. To check the accuracy of the reduced order models and the influence of the simplifying hypotheses on the results, the same problem is also analyzed using the finite element method. Excellent agreement is observed.     

Some problems in the antiplane shear deformation of bi-material wedges

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.     

环形薄板轴对称非线性屈曲的样条函数解法

环形薄板的大挠度计算因为边界条件复杂,仅有少数特殊情形的数值解答。均布边缘径向力作用下环形薄板非线性屈曲迄今尚未有研究成果。作者以三次B样条函数为试函数,用配点法计算环形薄板的大挠度。在12种不同的边界条件下,首次计算了环形薄板的压曲临界荷载及超临界荷载作用的变形。在所有的算例中均取得了收敛的数值结果。结果表明样条配点法具有收敛范围大、精度高和计算时间少的优点。     

环形薄板二维驻波的研究

在理论上和实验上对环形薄板二维驻波波节图形(克拉尼图形) 进行了研究. 通过在极坐标下对垂直板面方向小振动方程进行分离变量, 求解出环形薄板小振动方程在外边界悬空时分别在两种内边界条件, 即内边界悬空和内边界简支下的解析解的简正模式, 并计算了在第一种边条件下几种共振模式的径向波速近似值, 以及两种边条件下的圆形驻波波节线的半径和薄板的弹性模量. 发现通过调节环形薄板上点振动源的频率, 可精确控制薄板上出现的克拉尼图形. 实验上观察到了仅有圆形波节线, 仅有辐射状波节线, 以及两种波节线同时存在3 种简正模式的情形, 且波节线的数量可严格控制. 理论结果跟实验符合得很好.      

A note on the pure torsion of a circular cylinder for a compressible nonlinearly elastic material with nonconvex strain-energy

The large deformation torsion problem for an elastic circular cylinder subject to prescribed twisting moments at its ends is examined for a particular homogeneous isotropic compressible material, namely the Blatz-Ko material. For this material, the displacement equations of equilibrium in three-dimensional elastostatics can lose ellipticity at sufficiently large deformations. For the torsion problem, it is shown that this occurs when the prescribed torque reaches a critical value. For values of the twisting moment greater than this critical value, there is an axial core of the cylinder on which ellipticity holds, surrounded by an annular region where loss of ellipticity has occurred. The physical implications in terms of localized shear bands are briefly discussed.     

Large deflection problem of thin orthotropic circular plate with variable thickness under uniform pressure

Basic equations for large deflection theory of thin orthotropic circular plate with variable thickness are derived in this paper. The modified iteration method is adopted to solve the large deflection problem of thin orthotropic circular plate with variable thickness under uniform pressure. If ε=0, then the solution derived from the result in this paper coincides completely with the result given by J. Nowinski (using perturbation method) for solving large deflection problem of thin orthotropic circular plate with constant thickness under uniform pressure.     

A magnetoelectric screw dislocation interacting with a circular layered inclusion

Within the framework of the linear theory of magnetoelectroelasticity, the problem of a circular layered inclusion interacting with a generalized screw dislocation under remote anti-plane shear stress and in-plane magnetoelectric loads is investigated in this paper. The generalized dislocation can be located either in the matrix or in the circular layered inclusion. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different material properties. Using complex variable theory and the alternating technique, the solution of the present problem is expressed in terms of the solution of the corresponding homogeneous medium problem subjected to the same loading. Some numerical results are provided to investigate the influence of material combinations on the shear stress, electric field, magnetic and image force. These solutions can be used as Green's functions for the analysis of the corresponding magnetoelectric crack problem.     

Endliche Durchbiegungen beliebig eingespannter dünner Kreis- und Kreisringplatten im plastischen Materialbereich

Übersicht Endliche Durchbiegungen rotationssymmetrisch und sonst beliebig eingespannter Kreis- und Kreisringplatten werden untersucht. Die rotationssymmetrische Belastung sei so gewählt, daß in der Platte plastizierte Bereiche entstehen. Bei der Lösung dieses geometrisch und physikalisch nichtlinearen Problems wird eine Hypothese über die nichtlineare Spannungsverteilung postuliert und die Gültigkeit der Huber-Mises-Fließbedingung in den plastizierten Bereichen vorausgesetzt. Die Durchbiegung, die Verzerrungen und die Schnittlasten werden als Funktionen der radialen Laufkoordinate r, der radialen Verschiebung u(r) in der Mittelfläche, des radialen Biegewinkels (r), sowie der Ableitungen du/dr und d/dr bestimmt. Das gewonnene nichtlineare Differentialgleichungssystem ermöglicht die numerische Berechnung von u(r) und (r) für Kreis-und Kreisringplatten unter den genannten Voraussetzungen.

---

Summary Large deflections of axial-symmetrically but otherwise arbitrarily supported circular and annular plates are investigated. The loding is axially symmetrical and is extended in such a way that plastic ranges arise in the plate. To solve this geometrically and physically nonlinear problem, a nonlinear stress-hypothesis is assumed and Huber-Mises'yield condition is used. Deflection, stress, strain, membrane forces, and bending moments are determined as functions of the radial coordinate r, of the radial displacement u(r) in the middle-plane, of the radial bending angle (r) and of the derivatives du/dr and d/dr. From the resulting nonlinear system of two nonlinear differential equations numerical evaluations of u(r) and (r) for circular or annular plates under the above, mentioned conditions are obtained.

     

Some particular solutions for penny-shaped crack problem by using hypersingular integral equation or differential-integral equation

Summary A hypersingular integral equation or a differential-integral equation is used to solve the penny-shaped crack problem. It is found that, if a displacement jump (crack opening displacement COD) takes the form of (a ²−x ²−y ²)^1/2 x ^m y ⁿ , where a denotes the radius of the circular region, the relevant traction applied on the crack face can be evaluated in a closed form, and the stress intensity factor can be derived immediately. Finally, some particular solutions of the penny-shaped crack problem are presented in this paper. Received 1 July 1997; accepted for publication 13 October 1997     

General Nonsymmetric Postbuckling of Orthotropic Annular Plates

ABSTRACT

The general postbuckling behavior of orthotropic annular plates subjected to uniformly distributed forces is studied through the use of the Bubnov-Galerkin method. The transverse diplacement function is assumed to be in the form w = h if /, £ a,,p”+A-’-J cos m8 (=1.1=0 Wheref, = multipliers to be determined, a₁₀= 1, a₁₁(j # 0) = constants depending on the boundary conditions, p, θ = polar coordinates, and m = number of diametral nodal lines. Deflections, bending, and membrane stresses are presented in charts and listed in tables for 12 cases of boundary conditions, 4 loading cases, and various ratios of orthotropy.     

Natural Convection Induced by a Centrifugal Force Field in a Horizontal Annular Porous Layer Saturated with a Binary Fluid

The Darcy Model with the Boussinesq approximation is used to study natural convection in a horizontal annular porous layer filled with a binary fluid, under the influence of a centrifugal force field. Neumann boundary conditions for temperature and concentration are applied on the inner and outer boundary of the enclosure. The governing parameters for the problem are the Rayleigh number, Ra, the Lewis number, Le, the buoyancy ratio, j{\varphi } , the radius ratio of the cavity, R, the normalized porosity, e{\varepsilon } , and parameter a defining double-diffusive convection (a = 0) or Soret induced convection (a = 1). For convection in a thin annular layer (R → 1), analytical solutions for the stream function, temperature and concentration fields are obtained using a concentric flow approximation and an integral form of the energy equation. The critical Rayleigh number for the onset of supercritical convection is predicted explicitly by the present model. Also, results are obtained from the analytical model for finite amplitude convection for which the flow and heat and mass transfer are presented in terms of the governing parameters of the problem. Numerical solutions of the full governing equations are obtained for a wide range of the governing parameters. A good agreement is observed between the analytical model and the numerical simulations.     

The perturbation parameter in the problem of large deflection of clamped circular plates

In the problems of large deflection of clamped circular plates under uniformly distributed loads, various perturbation parameters relating to load, deflection, slope of deflection, membrane force, etc, are studied. For a general perturbation parameter, the variational principle is used for the solution of such a problem. The applicable range of these perturbation parameters are studied in detail. In the case of uniformly loaded plate, perturbation parameter relating to central deflection seems to be the best among all others. The method of determination of perturbation solution by means of variational principle can be used to treat a variety of problems, including the large deflection problems under combine loads. This paper is completed under the guidance of Prof. Chien Wei-zang.     

On state-space elastostatics within a plane stress sectorial domain – the wedge and the curved beam

The plane stress sectorial domain is analysed according to a state-space formulation of the linear theory of elasticity. When loading is applied to the straight radial edges (flanks), with the circular arcs free of traction, one has the curved beam; when loading is applied to the circular arcs, with the flanks free of traction, one has the elastic wedge. A complete treatment of just one problem (the elastic wedge, say) requires two state-space formulations; the first describes radial evolution for the transmission of the stress resultants (force and moment), while the second describes circumferential evolution for determination of the rates of decay of self-equilibrated loading on the circular arcs, as anticipated by Saint-Venant’s principle. These two formulations can be employed subsequently for the curved beam, where now radial evolution is employed for the Saint-Venant decay problem, and circumferential evolution for the transmission modes. Power-law radial dependence is employed for the wedge, and is quite adequate except for treatment of the so-called wedge paradox; for this, and the curved beam, the formulations are modified so that ln r takes the place of the radial coordinate r. The analysis is characterised by a preponderance of repeating eigenvalues for the transmission modes, and the state-space formulation allows a systematic approach for determination of the eigen- and principal vectors. The so-called wedge paradox is related to accidental eigenvalue degeneracy for a particular angle, and its resolution involves a principal vector describing the bending moment coupled to a decay eigenvector. Restrictions on repeating eigenvalues and possible Jordan canonical forms are developed. Finally, symplectic orthogonality relationships are derived from the reciprocal theorem.     

Analysis of anisotropic sector with a radial crack under anti-plane shear loading

In this paper, the anti-plane shear deformation of an anisotropic sector with a radial crack is investigated. The traction–traction boundary conditions are imposed on the radial edges and the traction-free condition is considered on the circular segment of the sector. A novel mathematical technique is employed for the solution of the problem. This technique consists of the use of some recently proposed finite complex transforms (Shahani, 1999), which have complex analogies to the standard finite Mellin transforms of the first and second kinds. However, it is essential to state the traction-free condition of the crack faces in the form of a singular integral equation which is done in this paper by describing an exact analytical method. The resultant dual integral equations are solved numerically to determine the stress intensity factors at the crack tips. In the special cases, the obtained results coincide with those cited in the literature.