Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

An exact analytical solution for freely vibrating piezoelectric coupled circular/annular thick plates using Reddy plate theory

Based on third-order shear deformation plate theory of Reddy, the authors aim to provide an exact analytical solution for free vibration analysis of thick circular/annular plates, both upper and lower surfaces of which are in contact with a piezoelectric layer. Natural frequencies are determined by the solution of the coupled electromechanical governing equations for a combination of free, soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the plate. The electrodes on each piezoelectric layer are assumed to be short-circuited. The Maxwell electrostatics equation is satisfied by adopting a half-sine distribution of the electric potential in the transverse direction of the piezoelectric layers. A comparison of the present exact natural frequencies for piezoelectric coupled circular/annular plates with different boundary conditions is made with previously published results obtained by the Mindlin plate theory and 3-D modified finite element method. The effects of plate parameters such as host thickness to radius ratios, inner to outer radius ratios and piezoelectric to host thickness ratios on the natural frequencies of laminated circular/annular plates are investigated for different combinations of boundary conditions. Results obtained by the present exact closed-form solutions can be served as benchmark data for investigators to validate their numerical and analytical methods in the future.     

2D Josephson junction

The paper is devoted to the determination of processes in the two-dimensional Josephson junction via numerical solution of 2+1 dimensional sine-Gordon equation (sGe) with suitably chosen boundary conditions. There is adopted an assumption that the normal derivative of the phase order parameter on a border of a junction is given by the surface current distribution on superconductive electrodes. This current distribution can be determined exactly since it makes a linear problem even in a presence of an external magnetic field although the surface current distribution is not uniform and can be singular. A rectangular junction is discussed as an example.     

Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations

This paper deals with free vibration analysis of radially functionally graded circular and annular sectorial thin plates of variable thickness, resting on the Pasternak elastic foundation. Differential quadrature method (DQM) is used to yield natural frequencies of the circular/annular sectorial plates under simply-supported and clamped boundary conditions on the basis of the classical plate theory (CPT). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is assumed to remain constant. The validity of the present solution is first examined by studying the convergence of the frequency parameters. Then, a comparison of results with those available in literature confirms the excellent accuracy of the present approach. Afterwards, the frequency parameters of the circular/annular sectorial thin plates with uniform, linear, and quadratic variations in thickness are computed for different boundary conditions and various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Intergranular fluxons in high-T c superconductors

The granular high-T _c superconductors can carry very low transport current, in comparison to that found in the bulk of the material. Magnetization and critical current measurements at very low field indicate that this low transport current behaves as expected from a critical state model. The presence of weak links between the grains in granular aggregates is firmly established, together with the Josephson character of such links. The existence of some kind of magnetic particles, and of a mechanism of pinning for them, is required to explain the critical state regime. In this paper we examine the flux structures which can be present in the granular systems, which can be described by an array of SQUIDs, and we show their similarities to fluxons in a continuous medium and to fluxons in uniform Josephson junctions. A simple model, based on a two-dimensional network of pointlike Josepson junctions, is adequate to demonstrate the existance of the IF (intergranular fluxon). Its characteristics depend on a single parameter, which gives the coupling strength between grains. The discreteness of the system is the cause of an intrinsic pinning of the intergranular fluxons.     

Elasticity solution of functionally graded circular and annular plates integrated with sensor and actuator layers using differential quadrature

Based on three-dimensional theory of elasticity axisymmetric static analysis of functionally graded circular and annular plates imbedded in piezoelectric layers is investigated using differential quadrature method (DQM). The plate has various edges boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. This method can give an analytical solution along the graded direction using the state space method (SSM) and an effective approximate solution along the radial direction using the one-dimensional DQM. The method is validated by comparing numerical results with the results obtained in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers on the mechanical behavior of plate is studied. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the static behavior of FG circular and annular plates are investigated.     

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

Intergranular fluxons in high-Tcsuperconductors

The granular high-T _c superconductors can carry very low transport current, in comparison to that found in the bulk of the material. Magnetization and critical current measurements at very low field indicate that this low transport current behaves as expected from a critical state model. The presence of weak links between the grains in granular aggregates is firmly established, together with the Josephson character of such links. The existence of some kind of magnetic particles, and of a mechanism of pinning for them, is required to explain the critical state regime. In this paper we examine the flux structures which can be present in the granular systems, which can be described by an array of SQUIDs, and we show their similarities to fluxons in a continuous medium and to fluxons in uniform Josephson junctions. A simple model, based on a two-dimensional network of pointlike Josepson junctions, is adequate to demonstrate the existance of the IF (intergranular fluxon). Its characteristics depend on a single parameter, which gives the coupling strength between grains. The discreteness of the system is the cause of an intrinsic pinning of the intergranular fluxons.     

Fluxon Centering in Josephson Junctions with Exponentially Varying Width

Nonlinear eigenvalue problems for fluxons in long Josephson junctions with exponentially varying width are treated. Appropriate algorithms are created and realized numerically. The results obtained concern the stability of the fluxons, the centering both magnetic field and current for the magnetic flux quanta in the Josephson junction as well as the ascertaining of the impact of the geometric and physical parameters on these quantities. Each static solution of the nonlinear boundary-value problem is identified as stable or unstable in dependence on the eigenvalues of associated Sturm-Liouville problem. The above compound problem is linearized and solved by using of the reliable Continuous analogue of Newton method.     

Zernike-gauss polynomials and optical aberrations of systems with gaussian pupils

In the first two Notes of this series,(l,2) we discussed Zernike circle and annular polynomials that represent optimally balanced classical aberrations of systems with uniform circular or annular pupils, respectively. Here we discuss Zernike-Gauss polynomials which are the corresponding polynomials for systems with Gaussian circular or annular pupils.(3-5) Such pupils, called apodized pupils, are used in optical imaging to reduce the secondary rings of the pointspread functions of uniform pupils.(6) Propagation of Gaussian laser beams also involves such pupils.     

Buckling of circular/annular Mindlin nanoplates via nonlocal elasticity

The present study proposes an analytical solution for the axisymmetric/asymmetric buckling analysis of moderately thick circular/annular Mindlin nanoplates under uniform radial compressive in-plane load. In order to consider small-scale effects, nonlocal elasticity theory of Eringen is employed. To ensure the efficiency and stability of the present methodology, the results are compared with other ones presented in the literature. Further the exact closed-form solution is obtained using three potential functions. In addition, the effect of small scales on buckling loads for different parameters such as geometry of the nanoplate, boundary conditions, and axisymmetric/asymmetric mode numbers, is investigated. It is observed that the buckling mode shape for annular nanoplates, which corresponds to the lowest critical buckling load, may be axisymmetric or asymmetric depending on boundary conditions, inner to outer radius ratios, and thickness of the nanoplate. In other words, for stiffer boundary conditions and smaller inner to outer radius ratios, the mode shape corresponding to the lowest critical buckling load is an asymmetric mode. Also, the difference between axisymmetric and asymmetric buckling loads for higher mode numbers, greater thickness to outer radius ratios and smaller outer radii decreases by increasing the nonlocal parameter.     

Deformation and tearing of circular plates with varying support conditions under uniform impulsive loads

Transient dynamic finite element analysis of circular plates with varying support configurations under uniform single square wave form impulsive load has been carried out in FEA package ANSYS. Experimental results of Teeling-Smith and Nurick [The deformation and tearing of thin circular plates subjected to impulsive loads. Int J Impact Eng 1991;11(1):77–91] and Nurick et al. [Tearing of blast loaded plates with clamped boundary conditions. Int J Impact Eng 1996;18(7–8):803–27] for the onset of thinning and tearing at the boundary of clamped circular plates subjected to uniformly loaded air blasts have been used to compare and validate the numerical simulation and procedure. The Mode II failure with respect to clamped circular plates has been simulated using a rupture strain criteria. Mode III failure or plastic shear sliding, has been considered using a shear strain failure criteria as proposed by Wen and Jones for plates. A stiffness reduction scheme has been proposed to decide on the initiation and progression of tearing in conjunction with suitable failure model under Modes II and III. The evolution of deflections, plastic zones, rupture zones and failure modes under the blast loading conditions are found to match well with the experimental results. The validated numerical model has further been used to study the effect of plate thickness on the deformation and tearing response of the circular plates subjected to impulsive loads. The deformation, tearing and shock absorption response of clamped circular plates under uniform impulsive loads with ring support of varying edge configurations at the boundary have also been numerically studied. Further, the response of circular plate–tube combination with varying boundary support configurations has been studied. The plate has been considered at the mid-span of the tube of length equal to the plate diameter with the ends of the tube modelled as clamped. The numerical model has been used to study the effect of tube thickness variations on the deformation and tearing response of the circular plate under shock loads. The response of tube–plate combinations under uniform impulsive loads with ring support at the plate–tube junction have also been numerically studied.     

On the null and nonzero fields for true and spurious eigenvalues of annular and confocal elliptical membranes

In this paper, the dual boundary element method (BEM) and the null-field boundary integral equation method (BIEM) are both employed to solve two-dimensional eigenproblems. The positions of true and spurious eigenvalues for circular, elliptical, annular and confocal elliptical membranes are analytically examined in the continuous system and numerically studied in the discrete system. To analytically study eigenproblems, the polar and elliptical coordinates in conjunction with the Bessel functions, the Mathieu functions, the Fourier series and eigenfunction expansions are adopted. The fundamental solution is expanded into the degenerate kernel while the boundary densities of circular and elliptical boundaries are expanded by using the Fourier series and eigenfunction expansion, respectively. Dirichlet and Neumann eigenproblems are both considered as well as simply and doubly-connected domains are both addressed. By employing the singular value decomposition (SVD) technique in the discrete system, the common right unitary vectors corresponding to the true eigenvalues for the singular and hypersingular formulations are found while the common left unitary vectors corresponding to the spurious eigenvalues are obtained for the singular formulation or hypersingular formulation. True eigenvalues depend on the boundary condition while spurious eigenvalues depend on the approach, the singular formulation or hypersingular formulation of BEM/BIEM. Nonzero field in the domain are analytically derived and are numerically verified in case of the true eigenvalue while the interior null field and nonzero field for the complementary domain are obtained in case of the spurious eigenvalue. Four examples, circular, elliptical, annular and confocal elliptical membranes, are considered to demonstrate the finding of the present paper. After comparing with the analytical and numerical results, good agreements are made. The dual BEM displays the dual structure in the unitary vector and the null field.     

Reduction of the Problem of Interaction of Cracks in a Restrained Half Space to Boundary Integral Equations

We propose an approach of boundary integral equations to the investigation of interaction of plane cracks under static loading in a half space with rigidly restrained surface. By the method of potential, we reduce the problem to the solution of a system of two-dimensional boundary integral equations of Newton-potential type for the unknown functions of displacements of the crack surfaces. For a circular crack perpendicular to the surface of the half space, we obtain the values of the stress intensity factors.     

功能梯度材料圆板的三维弹性分析

基于线弹性理论的基本方程,选用两个位移分量和两个应力分量作为状态变量,利用状态空间法建立了功能梯度材料轴对称圆板的三维状态方程.根据微分求积法,将状态方程在径向进行离散,考虑周边固支的边界条件,采用打靶法数值求解了材料常数沿板厚按幂率变化的轴对称弯曲问题和自由振动问题,为求解功能梯度材料三维弹性响应提供了一种方法.并且给出了功能梯度材料三维圆板的静动态响应受组分材料分布以及板厚径比变化的影响规律.     

Analytical layerwise free vibration analysis of circular/annular composite sandwich plates with auxetic cores

In the present research, free vibration of circular and annular sandwich plates with auxetic (negative Poisson’s ratio) cores and isotropic/orthotropic face sheets is investigated for different combinations of the boundary conditions. To ensure that the results are accurate and reliable, a global–local layerwise plate theory is employed instead of the traditional equivalent single-layer theories. The governing equations are derived based on Hamilton’s principle and solved using a Taylor transform whose center is located at the outer radius of the plate. Due to this hint, the resulting semi-analytical solution can be employed for both circular and annular sandwich plates. After investigation of vibration behavior of a single-layer annular auxetic plate, a comprehensive parametric study including evaluation of effects of the auxeticity for sandwich plates with isotropic and orthotropic face sheets, symmetric and asymmetric layups, different core to sheet thickness, radius to thickness, and inner to outer radius ratios, and various boundary conditions, is carried out. Results show that unlike the single-layer auxetic plates that exhibit a transition state, the auxeticity may considerably increase the natural frequencies and rigidities of the circular/annular sandwich plates, especially when the boundary conditions induce higher rigidity in the plate or when the fibers are along the radial direction. Accuracy of results of the employed layerwise theory and the proposed semi-analytical solution is verified by comparing the results with those of the three-dimensional theory of elasticity extracted from the ABAQUS software.     

A numerical procedure for multiple circular holes and elastic inclusions in a finite domain with a circular boundary

This paper describes a numerical procedure for solving two-dimensional elastostatics problems with multiple circular holes and elastic inclusions in a finite domain with a circular boundary. The inclusions may have arbitrary elastic properties, different from those of the matrix, and the holes may be traction free or loaded with uniform normal pressure. The loading can be applied on all or part of the finite external boundary. Complex potentials are expressed in the form of integrals of the tractions and displacements on the boundaries. The unknown boundary tractions and displacements are approximated by truncated complex Fourier series. A linear algebraic system is obtained by using Taylor series expansion without boundary discretization. The matrix of the linear system has diagonal submatrices on its diagonal, which allows the system to be effectively solved by using a block Gauss-Seidel iterative algorithm.     

Consideration of spatial variation of the length scale parameter in static and dynamic analyses of functionally graded annular and circular micro-plates

This article introduces new methods for static and free vibration analyses of functionally graded annular and circular micro-plates, which can take into account spatial variation of the length scale parameter. The underlying higher order continuum theory behind the proposed approaches is the modified couple stress theory. A unified way of expressing the displacement field is adopted so as to produce numerical results for three different plate theories, which are Kirchhoff plate theory (KPT), Mindlin plate theory (MPT), and third-order shear deformation theory (TSDT). Governing partial differential equations and corresponding boundary conditions are obtained following the variational approach and the Hamilton's principle. Derived systems of differential equations are solved numerically by utilizing the differential quadrature method (DQM). Comparisons to the results available in the literature demonstrate the high level of accuracy of the numerical results generated through the developed methods. Extensive analyses are presented in order to illustrate the influences of various geometric and material parameters upon static deformation profiles, stresses, and natural vibration frequencies. In particular, the length scale parameter ratio -which defines the length scale parameter variation profile-is shown to possess a profound impact on both static and dynamic behaviors of functionally graded annular and circular micro-plates.     

等强度环形板

本文利用小挠度环形薄板的初参数形式解和阶梯折算法,得到了任意边界条件下的等强度环形板的厚度分布计算公式。计算了一些常见边界条件下的实例。作了在相同的工况下,等厚度板和等强度板的体积对比。本文是文和的继续。     

Thermo elasticity solution of functionally graded,solid, circular,and annular plates integrated with piezoelectric layers using the differential quadrature method

Static analysis of functionally graded (FG) solid circular/annular plates imbedded in piezoelectric layers under thermo-electro mechanical load is investigated using the differential quadrature method. The plate has various edge boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. The method is validated by comparing numerical results with the results obtained in the literature. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the thermoelastic behavior of FG solid circular and annular plates are investigated.