Postbuckling of long unsymmetrically laminated composite plates under axial compression

An approximate solution is given for the postbuckling of infinitely long and unsymmetrically laminated composite plates. This solution is obtained by superposing a polynomial transverse displacement given by bending due to unsymmetric laminate configurations and a simple functional representation for the buckling mode in conjunction with the Galerkin method. Nondimensional parameters are used to express the approximate solution in a very simple and clear formulation. The results given by this solution for axial compression in the longitudinal direction are compared with the results given by the nonlinear finite element method (FEM) for finite length rectangular long plates. The influence of the boundary conditions on postbuckling response is also studied. For the FEM analysis, two different simply supported boundary conditions on the long edges of the plate are considered. It is found that these two sets of boundary conditions give different results for the buckling and postbuckling finite element analysis. In most cases the FEM analysis overestimate and, respectively, underestimate the approximate closed form solution, depending on the type of simply supported boundary condition considered. Thus, the approximate solution appears useful for design purposes as an averaged quantity between the two FEM analyses. Also, it is found that the reduced bending stiffness method can be successfully used for determining the approximate solution.     

Postbuckling of piezoelectric FGM plates subject to thermo-electro-mechanical loading

In this paper, we examine the postbuckling behavior of functionally graded material FGM rectangular plates that are integrated with surface-bonded piezoelectric actuators and are subjected to the combined action of uniform temperature change, in-plane forces, and constant applied actuator voltage. A Galerkin-differential quadrature iteration algorithm is proposed for solution of the non-linear partial differential governing equations. To account for the transverse shear strains, the Reddy higher-order shear deformation plate theory is employed. The bifurcation-type thermo-mechanical buckling of fully clamped plates, and the postbuckling behavior of plates with more general boundary conditions subject to various thermo-electro-mechanical loads, are discussed in detail. Parametric studies are also undertaken, and show the effects of applied actuator voltage, in-plane forces, volume fraction exponents, temperature change, and the character of boundary conditions on the buckling and postbuckling characteristics of the plates.     

Postbuckling of FGM plates with piezoelectric actuators under thermo-electro-mechanical loadings

A postbuckling analysis is presented for a simply supported, shear deformable functionally graded plate with piezoelectric actuators subjected to the combined action of mechanical, electrical and thermal loads. The temperature field considered is assumed to be of uniform distribution over the plate surface and through the plate thickness and the electric field considered only has non-zero-valued component E_Z. The material properties of functionally graded materials (FGMs) 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, and the material properties of both FGM and piezoelectric layers are assumed to be temperature-dependent. The governing equations are based on a higher order shear deformation plate theory that includes thermo-piezoelectric effects. The initial geometric imperfection of the plate is taken into account. Two cases of the in-plane boundary conditions are considered. A two step perturbation technique is employed to determine buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FGM plates with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, volume fraction distribution, applied voltage, the character of in-plane boundary conditions, as well as initial geometric imperfections are studied.     

Thermal,electrical, mechanical coupled mechanics for initial buckling analysis of smart plates and beams using discrete layer kinematics

Non-linear behavior of smart structures is of interest to researchers due to the possibilities for the elaboration of more effective actuators and sensors based on piezoelectric materials. The aim of the present work, is to present an integrated approach for the buckling behavior of smart beams and plates under multiple loading conditions. In order to present an accurate analysis, a coupled constitutive formulation between thermal, electrical and mechanical fields is elaborated incorporating non-linearity due to large displacements. An 8-node plate element was implemented in combination with discrete layer kinematics (LW) for the through-the-thickness representation of the structure. The issues of the critical buckling load under different electrical conditions as well as thermal and electrical loading are also presented. Experimental results contribute to the verification of the accuracy of the numerical analysis results and of the coupling mechanics in general.     

Thermo-piezoelectric effects on the postbuckling of axially-loaded hybrid laminated cylindrical panels

IntroductionCompositelaminatedcylindricalpanelhasbeenusedextensivelyasastructuralconfiguration,mainlyintheaerospaceindustry .Oneoftherecentadvancesinmaterialandstructuralengineeringisinthefieldofsmartstructureswhichincorporatesadaptivematerials.Bytakingadvantageofthedirectandconversepiezoelectriceffects,piezoelectriccompositestructurescancombinethetraditionalperformanceadvantagesofcompositelaminatesalongwiththeinherentcapabilityofpiezoelectricmaterialstoadapttotheircurrentenvironment.Therefore…     

Postbuckling of imperfect rectangular composite plates under inplane shear closed-form approximate solutions

In this paper, the postbuckling behavior of rectangular orthotropic laminated composite plates with initial imperfections under inplane shear load is investigated in a closed-form analytical manner. The plates under consideration are assumed to be infinitely long in the longitudinal direction. At the longitudinal edges, two different sets of boundary conditions are considered, specifically 1) simply supported edges and 2) fully clamped edges. Using Timoshenko-type shape functions for both the initial bifurcational buckling analysis and the subsequent Marguerre-type postbuckling studies, closed-form analytical solutions for the buckling loads and for the postbuckling state variables are derived. A comparison with geometrically non-linear finite element computations shows that the derived analysis approaches are suitable for postbuckling studies in load ranges not too far beyond bifurcational buckling as they are currently relevant for e.g., composite airframe structural analysis and design. Due to their strictly closed-form analytical nature, the presented analysis methods can be used conveniently in engineering practice for all application purposes where computational time is a crucial factor, especially for preliminary analysis and design or optimization procedures.     

压电板屈曲和后屈曲的有限元分析

采用板的一附剪切理论,并考虑结构的几何非线性,基于Ｔｏｔａｌ Ｌａｇｒａｎｇｅ方法,建立了在弱非线性耦合假设下压电智能的有限元控制方法,分析了没边界条件下压电板的屈曲和后屈曲,计算结果表明,在单向压力作用下,材料的压电效应和外加电压对板的屈曲载荷及后屈曲影响很小;而电场对位移加载简支板的屈曲影响较显著。该文为压电材料的工程应用提供了理论指导,同时提供了一种有效的有限元分析方法。     

压电介质有限变形的增量变分方程

从连续介质力学的基本理论出发，导出了压电介质几种非线性率型变分方程，在定义了各种增量之后，由率型变分方程得到了四种增量变分方程，即压电介质的ＴｏｔａｌＬａｇｒａｎｇｅ和ＵｐｄａｔｅｄＬａｇｒａｎｇｅ变分方程，它们是导出压电介质非线性有限元方程的基础，也可用于推导其它的简化理论。     

Distributed piezoelectric actuation of a bistable buckled beam

Bistable structures, such as buckled beams or plates, are characterized by a two-well potential. Their nonlinear properties are currently exploited in actuators design (e.g. MEMS micropumps, switches, memory cells) to produce relatively high displacements and forces with low actuation energies. We investigate the use of distributed multiparameter actuation to control the buckling and postbuckling behavior of a three-layer piezoelectric beam pinned at either end. A two-parameter bending actuation controls the transversal motion, whilst an axial actuation and a beam end-shortening modulate the tangent bending stiffness. The postbuckling behavior is studied by reducing to a 2 dof system a nonlinear extensible elastica model. When the bending actuation is spatially symmetric, the postbuckling phenomena are analogue to those obtained for a transversal midspan force, being characterized by a snap-through instability. The use of a two-parameter actuation opens new transition scenarios, where it is possible to get true quasi-static transitions between the two specular equilibria of the buckled beam, without any instability phenomenon. The efficiencies of these different transition paths are discussed in terms of energetic requirements and stability properties. A numerical example shows the technical feasibility of the proposed actuation technique.     

Basic equations of the theory of thermoviscoelastic plates with distributed sensors and actuators

The basic equations of the theory of thermoviscoelastic thin-walled plates with piezoelectric sensors and actuators under monoharmonic mechanical and electric loading are derived using the Kirchhoff–Love hypotheses. The thermomechanical behavior of passive and piezoactive materials is described using the concept of complex characteristics. Methods of solving nonlinear problems of active damping of thermomechanical vibrations of plates with sensors and actuators are considered. The effect of dissipative heating on the damping of axisymmetric vibrations of a thermoviscoelastic solid circular plate is analyzed as an example     

带旋转自由度C^0类任意四边形板（壳）单元

基于Ｒｅｉｓｓｎｅｒ－Ｍｉｎｄｉｌｉｎ板弯曲理论和Ｖｏｎ－Ｋａｒｍａｎ大挠度理论,采用单元域内和边界位移插值一致性的概念,将四节点等参弯曲单元与Ａｌｌｍａｎ膜变形二次插值模式相结合,对层合板壳的大挠度分析提供了一种实用的带旋转自由度的四节点Ｃ＾０类板单元。大量算例表明：该单元对板壳结构的线性强度、稳定性和后屈曲分析都表现出良好的收敛性和足够的工程精度。     

非线性弹性基础上矩形板热后屈曲分析

给出非线性弹性基础上矩形板在均匀和非均匀（抛物型）热分布作用下的后屈曲分析。采用摄动——Ｇａｌｅｒｋｉｎ混合法给出完善和非完善矩形板热屈曲载荷和热后屈曲平衡路径。数值计算结果表明，非线性弹性基础上矩形板具有不稳定的热后屈曲平衡路径，且对初始几何缺陷是敏感的     

Postbuckling analysis of axially-loaded laminated cylindrical shells with piezoelectric actuators

A compressive postbuckling analysis is presented for a laminated cylindrical shell with piezoelectric actuators subjected to the combined action of mechanical, electric and thermal loads. The temperature field considered is assumed to be a uniform distribution over the shell surface and through the shell thickness, and the electric field is assumed to be the transverse component E_Z only. The material properties are assumed to be independent of the temperature and the electric field. The governing equations are based on the classical shell theory with von Kármán–Donnell-type kinematic nonlinearity. The nonlinear prebuckling deformations and initial geometric imperfections of the shell are both taken into account. A boundary layer theory of shell buckling, which includes the effects of nonlinear prebuckling deformations, large deflections in the postbuckling range, and initial geometric imperfections of the shell, is extended to the case of hybrid laminated cylindrical shells. A singular perturbation technique is employed to determine the buckling loads and postbuckling equilibrium paths. The numerical illustrations concern the compressive postbuckling behavior of perfect and imperfect, cross-ply laminated cylindrical thin shells with fully covered or embedded piezoelectric actuators under different sets of thermal and electric loading conditions. The effects played by temperature rise, applied voltage, shell geometric parameter, stacking sequence, as well as initial geometric imperfections are studied.     

Variational principles of nonlinear piezothermoelastic media

Through the phenomenological approach, the nonlinear constitutive equations coupling the electro/magnetic thermoelastic media are derived. Several nonlinear variational principles for piezothermoelastic continua are presented and employed to formulate the incremental variational principles which are of important significance in practical applications such as the nonlinear finite element analysis, the buckling, postbuckling and dynamic stability analyses of piezoelectric smart structures. The work is supported by the National Natural Science Foundation, the Doctoral Education Foundation and the Aerospace Foundation.     

Thermomechanical Postbuckling of Imperfect Moderately Thick Plates on Nonlinear Elastic Foundations*

ABSTRACT

A postbuckling analysis is presented for a moderately thick rectangular plate subjected to combined axial compression and uniform temperature loading, and resting on a softening nonlinear elastic foundation. The cases of (1) thermal postbuckling of initially compressed plates and (2) compressive post-buckling of initially heated plates are considered. The initial geometrical imperfections of the plates are taken into account. Formulations are based on Reissner-Mindlin plate theory, considering first-order shear deformation effects, and including plate-foundation interaction and thermal effects. The analysis uses a deflection-type perturbation technique to determine buckling loads and postbuckling equilibrium paths. Numerical examples include the performance of perfect and imperfect, moderately thick plates resting on softening nonlinear elastic foundations. Typical results are presented in dimensionless graphical form.     

NONLINEAR BUCKLING BEHAVIOR OF DAMAGED COMPOSITE SANDWICH PLATES CONSIDERING THE EFFECT OF TEMPERATURE-DEPENDENT THERMAL AND MECHANICAL PROPERTIES

On the basis of the first-order shear deformation plate theory and the zig-zag deformation assumption, an incremental finite element formulation for nonlinear buckling analysis of the composite sandwich plate is deduced and the temperature-dependent thermal and mechanical properties of composite is considered. A finite element method for thermal or thermo-mechanical coupling nonlinear buckling analysis of the composite sandwich plate with an interfacial crack damage between face and core is also developed. Numerical results and discussions concerning some typical examples show that the effects of the variation of the thermal and mechanical properties with temperature, external compressive loading, size of the damage zone and ply angle of the faces on the thermal buckling behavior are significant. Project supported by the National Natural Science Foundation of China (No. 59975013).     

Buckling and post-buckling of laminated composite plates

This paper addresses the buckling and post-buckling of laminated composite plates using higher order shear deformation theory associated with Green–Lagrange non-linear strain–displacement relationships. All higher order terms arising from nonlinear strain–displacement relations are included in the formulation. The present plate theory satisfies zero transverse shear strain conditions at the top and bottom surfaces of the plate in von Karman sense. A C⁰ isoparametric finite element is developed for the present nonlinear model.     

具有分层损伤的不同加筋形式复合材料层合板的后屈曲性态研究

基于板的一阶剪切理论和V on-K arm an大挠度理论,分别推导了复合材料层合板和层合梁的几何非线性有限元列式,提出了含嵌入分层的复合材料加筋层合板在受压缩载荷作用下的后屈曲有限元分析方法,对在板厚方向具有不同分层位置的加筋板结构进行了有限元数值分析,研究了不同的加筋方式及筋的分布对具有分层损伤的复合材料加筋层合板的后屈曲性态的影响,所得结果对确定在压缩载荷作用下含损伤复合材料加筋层合板的剩余承载能力具有参考价值。     

Low-energy impact of adaptive cylindrical piezoelectric–composite shells

A theoretical framework for analyzing low-energy impacts of laminated shells with active and sensory piezoelectric layers is presented, including impactor dynamics and contact law. The formulation encompasses a coupled piezoelectric shell theory mixing first order shear displacement assumptions and layerwise variation of electric potential. An exact in-plane Ritz solution for the impact of open cylindrical piezoelectric–composite shells is developed and solved numerically using an explicit time integration scheme. The active impact control problem of adaptive cylindrical shells with distributed curved piezoelectric actuators is addressed. The cases of optimized state feedback controllers and output feedback controllers using piezoelectric sensors are analyzed. Numerical results quantify the impact response of cylindrical shells of various curvatures including the signal of curved piezoelectric sensors. Additional numerical studies quantify the impact response of adaptive cylindrical panels and investigate the feasibility of actively reducing the impact force.     

Nonlinear dynamic response and fatigue damage evolution for piezoelectric laminated plates with matrix cracks

Based on the Talreja’s damage model with tensor valued internal state variables, considering the effects of piezoelectricity, geometrical nonlinearity and the constitutive relations of the composite uni-ply plate with matrix cracks, the nonlinear dynamic equations of piezoelectric laminated plates with damage are established and the problems are solved by using the finite difference method. In numerical calculating examples, the nonlinear dynamics responses and the fatigue damage evolution curves of the plates are obtained, and the effects of piezoelectric materials, damage and external load on the nonlinear dynamic response and fatigue damage evolution of the plates are discussed.