Analytical solution for the cylindrical bending vibration of piezoelectric composite plates

An analytical solution for the cylindrical bending vibrations of linear piezoelectric laminated plates is obtained by extending the Stroh formalism to the generalized plane strain vibrations of piezoelectric materials. The laminated plate consists of homogeneous elastic or piezoelectric laminae of arbitrary thickness and width. Fourier basis functions for the mechanical displacements and electric potential that identically satisfy the equations of motion and the charge equation of electrostatics are used to solve boundary value problems via the superposition principle. The coefficients in the infinite series solution are determined from the boundary conditions at the edges and continuity conditions at the interfaces between laminae, which are satisfied in the sense of Fourier series. The formulation admits different boundary conditions at the edges of the laminated piezoelectric composite plate. Results for laminated elastic plates with either distributed or segmented piezoelectric actuators are presented for different sets of boundary conditions at the edges.     

Wave propagation in magneto-electro-elastic multilayered plates

An analytical treatment is presented for the propagation of harmonic waves in magneto-electro-elastic multilayered plates, where the general anisotropic and three-phase coupled constitutive equations are used. The state-vector approach is employed to derive the propagator matrix which connects the field variables at the upper interface to those at the lower interface of each layer. The global propagator matrix is obtained by propagating the solution in each layer from the bottom of the layered plate to the top using the continuity conditions of the field variables across the interfaces. From the global propagator matrix, we finally obtain the dispersion relation by imposing the traction-free boundary condition on the top and bottom surfaces of the layered plate. Dispersion curves, modal shapes, and natural frequencies are presented for layered plates made of orthotropic elastic (graphite–epoxy), transversely isotropic PZT-5A, piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄ materials. While the numerical results show clearly the influence of different stacking sequences as well as material properties on the field response, the general methodology presented in the paper could be useful to the analysis and design of layered composites made of smart piezoelectric and piezomagnetic materials.     

An exact solution for a multilayered two-dimensional decagonal quasicrystal plate

By extending the pseudo-Stroh formalism to two-dimensional decagonal quasicrystals, an exact closed-form solution for a simply supported and multilayered two-dimensional decagonal quasicrystal plate is derived in this paper. Based on the different relations between the periodic direction and the coordinate system of the plate, three internal structure cases for the two-dimensional quasicrystal layer are considered. The propagator matrix method is also introduced in order to treat efficiently and accurately the multilayered cases. The obtained exact closed-form solution has a concise and elegant expression. Two homogeneous quasicrystal plates and a sandwich plate made of a two-dimensional quasicrystal and a crystal with two stacking sequences are investigated using the derived solution. Numerical results show that the differences of the periodic direction have strong influences on the stress and displacement components in the phonon and phason fields; different coupling constants between the phonon and phason fields will also cause differences in physical quantities; the stacking sequences of the multilayer plates can substantially influence all physical quantities. The exact closed-form solution should be of interest to the design of the two-dimensional quasicrystal homogeneous and laminated plates. The numerical results can also be employed to verify the accuracy of the solution by numerical methods, such as the finite element and difference methods, when analyzing laminated composites made of quasicrystals.     

Circumferential wave in magneto-electro-elastic functionally graded cylindrical curved plates

Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical and electromagnetic properties, have greatly applying promises. Based on Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of circumferential harmonic waves in piezoelectric-piezomagnetic FGM cylindrical curved plates. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The dispersion curves of the piezoelectric-piezomagnetic FGM cylindrical curved plate and the corresponding non-piezoelectric and non-piezomagnetic cylindrical curved plates are calculated to show the influences of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism. Finally, a cylindrical curved plate at a different ratio of radius to thickness is calculated to show the influence of the ratio on the piezoelectric effect and piezomagnetic effect.     

Three-dimensional solution of smart laminated anisotropic circular cylindrical shells with imperfect bonding

This paper presents an exact solution for a simply-supported and laminated anisotropic cylindrical shell strip with imperfect bonding at the off-axis elastic layer interfaces and with attached anisotropic piezoelectric actuator and sensor subjected to transverse loading. In this research, the imperfect interface conditions are described in terms of linear relations between the interface tractions in the normal and tangential directions, and the respective discontinuities in displacements. The solution for an elastic (or piezoelectric) layer of the smart laminated cylindrical shell strip is obtained in terms of the six-dimensional (or eight-dimensional) pseudo-Stroh formalism, solution for multilayered system is then derived based on the transfer matrix method. Finally, a numerical example is presented to demonstrate the effect of imperfect interface on the static response of the smart laminated cylindrical shell. The derived solutions can serve as benchmark results to assess various approximate shell theories and numerical methods.     

A Bending-Gradient model for thick plates. Part I: Theory

This is the first part of a two-part paper dedicated to a new plate theory for out-of-plane loaded thick plates where the static unknowns are those of the Kirchhoff–Love theory (3 in-plane stresses and 3 bending moments), to which six components are added representing the gradient of the bending moment. The new theory, called the Bending-Gradient plate theory is described in the present paper. It is an extension to arbitrarily layered plates of the Reissner–Mindlin plate theory which appears as a special case of the Bending-Gradient plate theory when the plate is homogeneous. However, we demonstrate also that, in the general case, the Bending-Gradient model cannot be reduced to a Reissner–Mindlin model. In part two (Lebée and Sab, 2011), the Bending-Gradient theory is applied to multilayered plates and its predictions are compared to those of the Reissner–Mindlin theory and to full 3D Pagano’s exact solutions. The main conclusion of the second part is that the Bending-Gradient gives good predictions of both deflection and shear stress distributions in any material configuration. Moreover, under some symmetry conditions, the Bending-Gradient model coincides with the second-order approximation of the exact solution as the slenderness ratio L/h goes to infinity.     

Accurate modelling of piezoelectric plates: single-layered plate

Summary  The paper presents an efficient two-dimensional approach to piezoelectric plates in the framework of linear theory of piezoelectricity. The approximation of the through-the-thickness variations accounts for the shear effects and a refinement of the electric potential. Using a variational formalism, electromechanically coupled plate equations are obtained for the generalized stress resultants as well as for the generalized electric inductions. The latter are deduced from the conservative electric charge equation, which plays a crucial role in the present model. Emphasis is placed on the boundary conditions at the plate faces. The model is used to examine some problems for cylindrical bending of a single simply supported plate. Number of situations are examined for a piezoelectric plate subject to (i) an applied electric potential, (ii) a surface density of force, and (iii) a surface density of electric charge. The through-thickness distributions of electromechanical quantities (displacements, stresses, electric potential and displacement) are obtained, and compared with results provided by finite element simulations and by a simplified plate model without shear effects. A good agreement is observed between the results coming from the present plate model and finite element computations, which ascertains the effectiveness of the proposed approach to piezoelectric plates. Received 17 July 2000; accepted for publication 26 September 2000     

Static deformation of a multilayered one-dimensional hexagonal quasicrystal plate with piezoelectric effect

Quasicrystals (QCs) are sensitive to the piezoelectric (PE) effect. This paper studies static deformation of a multilayered one-dimensional (1D) hexagonal QC plate with the PE effect. The exact closed-form solutions of the extended displacement and traction for a homogeneous piezoelectric quasicrystal (PQC) plate are derived from an eigensystem. The general solutions for multilayered PQC plates are then obtained using the propagator matrix method when mechanical and electrical loads are applied on the top surface of the plate. Numerical examples for several sandwich plates made up of PQC, PE, and QC materials are provided to show the effect of stacking sequence on phonon, phason, and electric fields under mechanical and electrical loads, which is useful in designing new composites for engineering structures.     

Propagation of axial shear magneto–electro-elastic waves in piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities

The integral equations of the scattering problem for piezoelectric–piezomagnetic composites with an inhomogeneity are derived. In the long-wave limit, the solutions of these integral equations for the composites containing a single inhomogeneous fiber are solved in close forms. The total scattering cross-section for the one-fiber composites is also obtained. By the so-called effective field method, the multi-fiber scattering problem is simplified to the one-fiber scattering problem, and the analytical expressions of magneto–electro-elastic fields for the multi-fiber composites are obtained in the long-wave limit. These solved magneto–electro-elastic fields are then used to solve the expressions of the static effective moduli, effective wave velocity and attenuation factor of piezoelectric–piezomagnetic composites with randomly distributed cylindrical inhomogeneities. Through numerical examples, it concludes that, if the random set of fiber cross-sections is homogeneous and isotropic, the effective field method is coincident with the Mori–Tanaka mean field method when the static effective moduli of piezoelectric–piezomagnetic composites are looked for. Moreover, the rules of the effective wave velocity versus the volume fraction of fibers are investigated for specific materials.     

Asymptotic splitting in the three-dimensional problem of elasticity for non-homogeneous piezoelectric plates

A novel asymptotic approach to the theory of non-homogeneous anisotropic plates is suggested. For the problem of linear static deformations we consider solutions, which are slowly varying in the plane of the plate in comparison to the thickness direction. A small parameter is introduced in the general equations of the theory of elasticity. According to the procedure of asymptotic splitting, the principal terms of the series expansion of the solution are determined from the conditions of solvability for the minor terms. Three-dimensional conditions of compatibility make the analysis more efficient and straightforward. We obtain the system of equations of classical Kirchhoff's plate theory, including the balance equations, compatibility conditions, elastic relations and kinematic relations between the displacements and strain measures. Subsequent analysis of the edge layer near the contour of the plate is required in order to satisfy the remaining boundary conditions of the three-dimensional problem. Matching of the asymptotic expansions of the solution in the edge layer and inside the domain provides four classical plate boundary conditions. Additional effects, like electromechanical coupling for piezoelectric plates, can easily be incorporated into the model due to the modular structure of the analysis. The results of the paper constitute a sound basis to the equations of the theory of classical plates with piezoelectric effects, and provide a trustworthy algorithm for computation of the stressed state in the three-dimensional problem. Numerical and analytical studies of a sample electromechanical problem demonstrate the asymptotic nature of the present theory.     

Axisymmetric bending analysis of thick functionally graded circular plates using fourth-order shear deformation theory

In the present article, axisymmetric bending and stretching of functionally graded (FG) circular plates subjected to uniform transverse loading based on fourth-order shear deformation plate theory (FOST) have been studied. Using a fourth-order shear deformation theory, the solutions for deflection and rotation functions of FG plates are presented in terms of the corresponding quantities for a homogeneous plate using the classical plate theory (CPT), from which solutions one can easily obtain the FOST solutions for axisymmetric bending of FG circular plates. It is assumed that the effective mechanical properties of the functionally graded plates through the thickness are continuous functions of the volume fractions of the constituent parts which are themselves defined by a power-law function. Numerical results for maximum deflection and shear stress are presented for various percentages of ceramic–metal volume fractions. These results are also compared with those obtained from the first-order shear deformation plate theory of Mindlin (FST), the third-order shear deformation plate theory of Reddy (TST) as well as the exact three-dimensional elasticity solution. It is found that although the maximum deflections obtained using FOST and TST are close to each other, the through-thickness shear stress is predicted more accurately by the FOST formulation than by the TST.     

多层横观各向同性圆柱壳的三维弹性理论解

本文从三维弹性理论出发,用特征函数法研究多层横观各向同性圆柱壳的轴对称问题.把位移和应力分量的齐次解表达成特征函数展开式,并把特解部分用Fourier级数表示.以多层圆柱壳的内、外柱面作为齐次边界,同时考虑层间的连续条件,推导出问题的特征方程并用Muller法求解.文中运用传递矩阵技术处理多层问题,并用边界型最小二乘配点法处理端部边界条件.作为实例,对双层圆柱壳作了数值计算.     

Stability of viscoelastic rectangular plate with a piezoelectric layer subjected to follower force

The effects of a piezoelectric layer on the stability of viscoelastic plates subjected to the follower forces are evaluated. The differential equation of motion of the viscoelastic plate with the piezoelectric layer is formulated using the two-dimensional viscoelastic differential constitutive relation and the thin plate theory. The weak integral form of the differential equations and the force boundary conditions are obtained. Using the element-free Galerkin method, the governing equation of the viscoelastic rectangular plate with elastic dilatation and Kelvin–Voigt distortion is derived, subjected to the follower forces coupled with the piezoelectric effect. A generalized complex eigenvalue problem is solved, and the force excited by the piezoelectric layer due to external voltage is modeled as the follower tensile force; this force is used to improve the stability of the non-conservative viscoelastic plate. For the viscoelastic plate with various boundary conditions, the results for the instability type and the critical loads are presented to show the variations in these factors with respect to the location of the piezoelectric layers and the applied voltages. The stability of the viscoelastic plates can be effectively improved by the determination of the optimal location for the piezoelectric layers and the most favorable voltage assignment.     

面内变刚度薄板弯曲问题的挠度?弯矩耦合神经网络方法

发展了一种求解面内变刚度功能梯度薄板弯曲问题的神经网络方法. 面内变刚度薄板弯曲问题的偏微分控制方程为一复杂的4阶偏微分方程, 传统的基于强形式的神经网络解法在求解该偏微分方程时可能会遇到难以收敛、边界条件难以处理的情况. 本文基于Kirchhoff薄板弯曲理论, 提出了一种直角坐标系下任意面内变刚度薄板弯曲问题的神经网络解法. 神经网络模型包含挠度网络与弯矩网络, 分别用于预测薄板的挠度与弯矩, 从而将求解4阶偏微分方程转换为求解一系列二阶偏微分方程组, 通过对挠度、弯矩试函数的构造可使得神经网络计算结果严格满足边界条件. 在误差的反向传播中, 根据本文提出的误差函数公式计算训练误差并结合Adam优化算法更新模型的内部参数. 求解了不同边界条件、形状的面内变刚度薄板弯曲问题, 并将所得计算结果与理论解、有限元解进行对比. 研究表明, 本文模型对于求解面内变刚度薄板弯曲问题具备适应性, 虽然模型中的弯矩网络收敛较挠度网络要慢, 但本文方法在试函数的构造上更为简单、适应性更强.      

Buckling analysis of sandwich plates with FGM face sheets resting on elastic foundation with various boundary conditions: an analytical approach

This paper presents an analytical investigation on the buckling analysis of symmetric sandwich plates with functionally graded material (FGM) face sheets resting on an elastic foundation based on the first-order shear deformation plate theory (FSDT) and subjected to mechanical, thermal and thermo-mechanical loads. The material properties of FGM face sheets are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic material. An analytical approach is used to reduce the governing equations of stability and then solved using an analytical solution which is named as power series Frobenius method for symmetric sandwich plates with six different boundary conditions. A detailed numerical study is carried out to examine the influence of the plate aspect ratio, side-to-thickness ratio, loading type, sandwich plate type, volume fraction index, elastic foundation coefficients and boundary conditions on the buckling response of FGM sandwich plates. This has not been done before and serves to fill the gap of knowledge in this area.     

A comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

Green’s functions for transversely isotropic piezoelectric functionally graded multilayered half spaces

By virtue of two systems of vector functions and the propagator matrix method, Green’s functions for transversely isotropic, piezoelectric functionally graded (exponentially graded in the vertical direction), and multilayered half spaces are derived. It is observed that the homogeneous solution and propagator matrices for each functionally graded layer in the transformed domain are independent of the choice of the two systems of vector functions. For a point force and point charge density applied at any location of the functionally graded half space, Green’s functions are expressed in terms of one-dimensional infinite integrals. To carry out the numerical integral involving Bessel functions, an adaptive Gauss quadrature approach is introduced and modified. The piezoelectric functionally graded Green’s functions include those in the corresponding elastic functionally graded media as special results with the latter being also unavailable in the literature. Two piezoelectric functionally graded half-space models are analyzed numerically: one is a functionally graded PZT-4 half space, and the other a coated functionally graded PZT-4 layer over a homogeneous BaTiO₃ half space. The effects of different exponential factors on Green’s function components are clearly demonstrated, which could be useful in the design and manufacturing of piezoelectric functionally graded structures.     

Analytical solutions for single- and multi-span functionally graded plates in cylindrical bending

A recently developed plate theory using the concept of shape function of the transverse coordinate parameter is extended to determine the stress distribution in an orthotropic functionally graded plate subjected to cylindrical bending. The transfer matrix method is presented to derive the shape function. The equations governing the plate deformation are then solved analytically using the transfer matrix method for arbitrary boundary conditions. For a simply supported functionally graded plate, a comparison of the present solution with the exact elasticity solution, the first- and third-order shear deformation plate theories is presented and discussed. It is demonstrated that the present method yields more accurate stresses than the first- and third-order shear deformation theories. The effect of boundary conditions and inhomogeneity of material on the displacements and stresses in functionally graded plates are investigated. A multi-span functionally graded plate with arbitrary boundary conditions is further considered to demonstrate the efficiency of the present method.     

Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的辛叠加解

研究Winkler地基上正交各向异性矩形薄板弯曲方程所对应的Hamilton正则方程, 计算出其对边滑支条件下相应Hamilton算子的本征值和本征函数系, 证明该本征函数系的辛正交性以及在Cauchy主值意义下的完备性, 进而给出对边滑支边界条件下Hamilton正则方程的通解, 之后利用辛叠加方法求出Winkler地基上四边自由正交各向异性矩形薄板弯曲问题的解析解. 最后通过两个具体算例验证了所得解析解的正确性.     

Interface edge crack in a multiferroic semicylinder

The purpose of the present work is to study the interface edge cracking problem in a semicircular cylindrical multiferroic composite theoretically by the methods of infinite series and singular integral equation. Numerical results of the stress intensity factor (SIF) are obtained and the computational accuracy is demonstrated. Discussion on the numerical results indicates that: (a) in order to reduce the SIF the inner piezoelectric layer should be softer and the outer piezomagnetic layer be stiffer; (b) the piezoelectric stiffening affect the SIF notably, but the piezomagnetic stiffening can only influence it very little.