Parametric instability of conical shells by the Generalized Differential Quadrature method

The parametric instability of truncated conical shells of uniform thickness under periodic edge loading is examined. The material considered is homogeneous and isotropic. This is the first instance that the Generalized Differential Quadrature (GDQ) method is used to study the effects of boundary conditions on the parametric instability in shells. The formulation is based on the dynamic version of Love's first approximation for thin shells. A formulation is presented which incorporates the GDQ method in the assumed‐mode method to reduce the partial differential equations of motion to a system of coupled Mathieu–Hill equations. The principal instability regions are then determined by Bolotin's method. Assumptions made in this study are the neglect of transverse shear deformation, rotary inertia as well as bending deformations before instability. Copyright © 1999 John Wiley & Sons, Ltd.     

Cylindrical bending vibration of functionally graded piezoelectric shells using the method of perturbation

Based on the three-dimensional (3D) piezoelectricity, two asymptotic formulations for the cylindrical bending vibration of simply supported, functionally graded (FG) piezoelectric cylindrical shells with open-circuit and closed-circuit surface conditions are presented. The normal electric displacement and electric potential are prescribed to be zero on the lateral surfaces. In the present asymptotic formulations the material properties are regarded to be heterogeneous through the thickness coordinate. Afterwards, they are further specified to be constant in single-layer shells, to be layerwise constant in multilayered shells and to obey an identical exponent-law distribution in FG shells. The method of multiple time scales is used to eliminate the secular terms arising from the regular asymptotic expansion. The orthonormality and solvability conditions for various orders are derived. The recursive property among the motion equations of various order problems is shown. The present asymptotic formulations are applied to several illustrative examples. The accuracy and the rate of convergence of the present asymptotic solutions are evaluated. The coupled electro–elastic effect and the influence of the material-property gradient index on the free-vibration behavior of FG piezoelectric shells are studied.     

2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution

In this paper, the Generalized Differential Quadrature (GDQ) method is applied to study the dynamic behaviour of laminated composite doubly-curved shells of revolution. The First-order Shear Deformation Theory (FSDT) is used to analyse the above mentioned moderately thick structural elements. In order to include the effect of the initial curvature a generalization of the Reissner–Mindlin theory, proposed by Toorani and Lakis, is adopted. The governing equations of motion, written in terms of stress resultants, are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The discretization of the system by means of the Differential Quadrature (DQ) technique leads to a standard linear eigenvalue problem, where two independent variables are involved. Results are obtained taking the meridional and circumferential co-ordinates into account, without using the Fourier modal expansion methodology. Comparisons between the Reissner–Mindlin and Toorani–Lakis theory are presented. Furthermore, GDQ results are compared with those presented in literature and the ones obtained by using commercial programs such as Abaqus, Ansys, Nastran, Straus and Pro/Mechanica. Very good agreement is observed.     

A finite element model for the analysis of 3D axisymmetric laminated shells with piezoelectric sensors and actuators

This paper presents the development of a semi-analytical axisymmetric shell finite element model with piezoelectric layers using the 3D linear elasticity theory. The piezoelectric effect of the material could be used as sensors and/or actuators in way to control shell deformation. In the present 3D axisymmetric model, the equations of motion are expressed by expanding the displacement field using Fourier series in the circumferential direction. Thus, the 3D elasticity equations of motion are reduced to 2D equations involving circumferential harmonics. In the finite element formulation the dependent variables, electric potential and loading are expanded in truncated Fourier series. Special emphasis is given to the coupling between symmetric and anti-symmetric terms for laminated materials with piezoelectric rings. Numerical results obtained with the present model are found to be in good agreement with other finite element solutions.     

Elliptically delaminated buckling near the surface of piezoelectric laminated shells under electric and thermal loads

This paper reports the result of an investigation into the local buckling for the delimitation of an elliptic shape near the surface of piezoelectric laminated shells, where local delaminated sub-shells may be monolayer and multiplayer, and the geometrical axes of elliptically delaminated sub-shells are inconsistent with the geometrical main axes of piezoelectric laminated base-shells. From example calculations, the effects of the geometrical and physical parameters, stacking models of piezoelectric laminated base-shells, elliptically laminated sub-shells, and coupled electric and thermal loads on the local buckling for elliptically delaminated sub-shells near the surface of piezoelectric laminated base-shells are described and discussed.     

Buckling analysis of cross-ply laminated conical panels using GDQ method

The buckling analysis of cross-ply laminated conical shell panels with simply supported boundary conditions at all edges and subjected to axial compression is studied. The conical shell panel is a very interesting problem as it can be considered as the general case for conical shells when the subtended angle is set to 2π and also cylindrical panels and shells when the semi-vertex angle is equal to zero. Equations were derived using classical shell theory of Donnell type and solved using generalized differential quadrature method. The results are compared and validated with the known results in the literature. The effects of subtended angle, semi-vertex angle, length, thickness and radius of the panel on the buckling load and mode are investigated.     

Coupled efficient layerwise and smeared third order theories for vibration of smart piezolaminated cylindrical shells

The improved third order zigzag theory and its smeared counterpart (without the zigzag effect), recently developed by the authors for static analysis of piezoelectric laminated cylindrical shells, are extended to dynamics. The piezoelectric layers are considered as radially polarized to make use of the extension actuation mechanism that is best suited for effective actuation and sensing. The zigzag theory accounts for the layerwise variation of inplane displacements and includes the transverse normal extensibility under electric field, and also satisfies the conditions on transverse shear stresses at the layer interfaces and at the inner and outer surfaces of the shell. Yet, the number of primary displacement variables is only five, same as its smeared counterpart. The two theories are critically assessed for their accuracy by direct comparison with the three dimensional piezoelasticity solutions for free and forced vibration response of simply supported smart angle-ply infinite-length and cross-ply finite-length shells, with a variety of heterogeneous composite and sandwich laminates. It is shown that the zigzag theory, in spite of being computationally efficient, is very accurate even for shells with highly inhomogeneous laminates. In contrast, the smeared third order theory is grossly inadequate for smart shells made of inhomogeneous composite and sandwich substrates.     

Solving anti-plane problems of piezoelectric materials by the Trefftz finite element approach

Applications of the Trefftz finite element method to anti-plane electroelastic problems are presented in this paper. A dual variational functional is constructed and used to derive Trefftz finite element formulation. Special trial functions which satisfy boundary conditions are also used to develop a special purpose element with local defects. The performance of the proposed element model is assessed by an example and comparison is made with results obtained by other approaches. The Trefftz finite element approach is demonstrated to be ideally suited for the analysis of the anti-plane problem.The work was performed under the auspices of an Australian Professorial Fellowship Program with grant number DP0209487 and 21st Century Education Promotion Key project from Tianjin University.     

压电复合材料三维壳体简化数值建模研究

为有效分析三维压电复合材料壳体结构非线性、 单向耦合压电弹性问题, 基于变分渐近方法(VAM)建立了壳体结构在机械和电场作用下的简化模型。推导了基于旋转张量分解概念的压电复合材料三维壳体能量表达式; 利用变分渐近法将三维壳体严格拆分为二维壳体线性分析和沿法线方向的一维非线性分析; 进行了降维后近似能量推导及Reissner-Mindlin形式转换; 提供了三维场重构关系以得到沿厚度方向的准确应力分布。通过对由4层压电复合材料构成的壳体柱形弯曲算例分析表明: 基于该理论和重构过程开发的变分渐近程序VAPAS重构生成的三维应力场精确性较一阶剪切变形理论和古典层合理论更好, 与三维有限元精确解相吻合, 表明该压电复合材料壳体模型的有效性。      

Material properties of piezoelectric composites by BEM and homogenization method

Applications of boundary element method (BEM) to piezoelectric composites in conjunction with homogenization approach for determining their effective material properties are discussed in this paper. The composites considered here consist of inclusion and matrix phases. The homogenization model for composites with inhomogeneities is developed and introduced into a BE formulation to provide an effective means for estimating overall material constants of two-phase composites. In this model, a representative volume element (RVE) is used whose volume average stress and strain are calculated by the boundary tractions and displacements of the RVE. Thus BEM is suitable for performing calculations on average stress and strain fields of the composites. Numerical results for a piezoelectric plate with circular inclusions are presented to illustrate the application of the proposed micromechanics––BE formulation.     

Numerical investigation of functionally graded cylindrical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery

A 2D Unconstrained Third Order Shear Deformation Theory (UTSDT) is presented for the evaluation of tangential and normal stresses in moderately thick functionally graded cylindrical shells subjected to mechanical loadings. Eight types of graded materials are investigated. The functionally graded material consists of ceramic and metallic constituents. A four parameter power law function is used. The UTSDT allows the presence of a finite transverse shear stress at the top and bottom surfaces of the graded cylindrical shell. In addition, the initial curvature effect included in the formulation leads to the generalization of the present theory (GUTSDT). The Generalized Differential Quadrature (GDQ) method is used to discretize the derivatives in the governing equations, the external boundary conditions and the compatibility conditions. Transverse and normal stresses are also calculated by integrating the three dimensional equations of equilibrium in the thickness direction. In this way, the six components of the stress tensor at a point of the cylindrical shell or panel can be given. The initial curvature effect and the role of the power law functions are shown for a wide range of functionally cylindrical shells under various loading and boundary conditions. Finally, numerical examples of the available literature are worked out.     

Residual stress in piezoelectric poly(vinylidene-fluoride-co-trifluoroethylene) thin films deposited on silicon substrates

The residual stress and its evolution with time in poly(vinylidene-fluoride-co-trifluoroethylene) (P(VDF-TrFE) (72/28)) piezoelectric polymer thin films deposited on silicon wafers were investigated using the wafer curvature method. Double-side polished silicon wafers with minimized initial wafer warpage were used to replace single-side polished silicon wafers to obtain significantly improved reliability for the measurement of the low residual stress in the P(VDF-TrFE) polymer thin films. Our measurement results showed that all the P(VDF-TrFE) films possessed a tensile residual stress, and the residual stress slowly decreased with time. Our analysis further indicates that the tensile stress could arise from the thermal mismatch between the P(VDF-TrFE) film and the silicon substrate. Besides possible viscoelastic creep mechanism in thermoplastic P(VDF-TrFE) films, microcracks with widths in the range of tens of nanometers appeared to release the tensile residual stress.     

Nonlinear vibrations of laminated circular cylindrical shells: Comparison of different shell theories

The geometrically nonlinear forced vibrations of laminated circular cylindrical shells are studied by using the Amabili–Reddy higher-order shear deformation theory. An energy approach based on Lagrange equations, retaining modal damping, is used in order to obtain the equations of motion. An harmonic point excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are studied by using the pseudo-arclength continuation method and bifurcation analysis. A one-to-one internal resonance is always present for a complete circular cylindrical shell, giving rise to pitchfork bifurcations of the nonlinear response with appearance of a second branch with travelling wave response and quasi-periodic vibrations. The numerical results obtained by using the Amabili–Reddy shell theory are compared to those obtained by using an higher-order shear deformation theory retaining only nonlinear term of von Kármán type and the Novozhilov classical shell theory.     

Analysis of isotropic and multilayered plates and shells by using a generalized higher-order shear deformation theory

This paper introduces a generalized 5 degrees of freedom (DOF) higher-order shear deformation theory (HSDT) to study the bending and free vibration of plates and shells, which may be used to create other HSDTs. It also introduces a new HSDT for shells that is more accurate than many available HSDTs despite having the same 5DOF, and which is also able to reproduce the well-known Soldatos’ HSDT as special case. The governing equations and boundary conditions of the generalized formulation are derived by employing the principle of virtual work. These equations are solved via Navier-type closed-form solutions. Static and dynamic results are presented for plates and cylindrical and spherical shells with simply supported boundary conditions. Panels are subjected to sinusoidal, distributed and point loads. Results are provided for thick to thin as well as shallow and deep shells. Results from the new and well-known HSDTs introduced and reproduced based on the present generalized 5DOF HSDT are compared with the exact three-dimensional elasticity solution. The present new HSDT for plates and shells is found to be more accurate than the well-known HSDTs developed by other authors, for analyzing the static and free vibration of isotropic and multilayered composite plates and shells.     

Vibration of axisymmetric composite piezoelectric shells coupled with internal fluid

This paper presents the theoretical and finite element formulations of piezoelectric composite shells of revolution filled with compressible fluid. The originality of this work lies (i) in the development of a variational formulation for the fully coupled fluid/piezoelectric structure system, and (ii) in the finite element implementation of an inexpensive and accurate axisymmetric adaptive laminated conical shell element. Various modal results are presented in order to validate and illustrate the efficiency of the proposed fluid–structure finite element formulation. Copyright © 2007 John Wiley & Sons, Ltd.     

Post buckling analysis of shear deformable shallow shells by the boundary element method

This paper presents four boundary element formulations for post buckling analysis of shear deformable shallow shells. The main differences between the formulations rely on the way non‐linear terms are treated and on the number of degrees of freedom in the domain. Boundary integral equations are obtained by coupling boundary element formulation of shear deformable plate and two‐dimensional plane stress elasticity. Four different sets of non‐linear integral equations are presented. Some domain integrals are treated directly with domain discretization whereas others are dealt indirectly with the dual reciprocity method. Each set of non‐linear boundary integral equations are solved using an incremental approach, where loads and prescribed boundary conditions are applied in small but finite increments. The resulting systems of equations are solved using a purely incremental technique and the Newton–Raphson technique with the Arc length method. Finally, the effect of imperfections (obtained from a linear buckling analysis) on the post‐buckling behaviour of axially compressed shallow shells is investigated. Results of several benchmark examples are compared with the published work and good agreement is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

Properties of piezoelectric actuator on silicon membrane, prepared by screen printing method

Characteristics of piezoelectric actuator on Si membrane were investigated. Si membranes were fabricated as a function of size using bulk micromachining method. Bottom electrode Ag–Pd and piezoelectric thick films were fabricated using screen printing method, respectively. Piezoelectric thick films were sintered by rapid thermal annealing (RTA). Top electrodes Pt were deposited by DC sputtering system. We analyzed micro structure by scanning electron microscope (SEM) and investigated dynamic properties by MTI2000. Therefore, piezoelectric thick film on Si membrane had P_r of 15.7 μC cm⁻². The maximum displacement of micro actuator had 0.05 μm. We find the combination of thick film printing and MEMS process to form a Si membrane micro actuator.     

Active damping of geometrically nonlinear vibrations of doubly curved laminated composite shells

This paper deals with the analysis of active constrained layer damping (ACLD) of geometrically nonlinear transient vibrations of doubly curved laminated composite shells. Vertically/obliquely reinforced 1–3 piezoelectric composite (PZC) and active fiber composite (AFC) materials are used as the materials of the constraining layer of theACLD treatment. The Golla–Hughes–McTavish (GHM) method has been implemented to model the constrained viscoelastic layer of the ACLD treatment in time domain. The first-order shear deformation theory (FSDT) and the Von Kármán type non-linear strain displacement relations are used for analyzing this coupled electro-elastic problem. A three dimensional finite element (FE) model of doubly curved laminated smart composite shells integrated with ACLD patches has been developed to investigate the performance of these patches for controlling the geometrically nonlinear transient vibrations of the shells. The numerical results indicate that the ACLD patches significantly improve the damping characteristics of the doubly curved laminated cross-ply and angle-ply shells for suppressing their geometrically nonlinear transient vibrations. It is found that the performance of the ACLD patch with its constraining layer being made of the AFC is significantly higher than that of the ACLD patch with vertically/obliquely reinforced 1–3 PZC constraining layer. The effects of variation of piezoelectric fiber orientation in both the obliquely reinforced 1–3 PZC and the AFC constraining layers on the control authority of the ACLD patches have also been investigated.     

Modelling and optimization of laminated adaptive shells of revolution

In this paper two shell finite element models are presented for the structural analysis of composite laminated piezoelectric shells. One is an axisymmetric conical frustum with two nodal rings and the other is a conic shell panel with eight nodes. Both models are based in a mixed laminated theory that combines a higher order shear deformation theory for the mechanical displacement field with a layerwise representation with linear functions for the electric potential through each piezoelectric layer. In order to obtain the optimal design sensitivities analysis and optimization techniques based in the nonlinear mathematical programming are used. The design objectives can be the minimization of the deformed structure or the maximization of the natural fundamental frequency and the design variables are the electric potential difference applied to the actuators or the ply thicknesses among others.