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.     

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).     

Thermal buckling of imperfect functionally graded plates

Thermal buckling analysis of rectangular functionally graded plates (FGPs) with geometrical imperfections is presented in this paper. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the classical plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading as uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect FGP. The results are reduced and compared with the results of perfect functionally graded and imperfect isotropic plates.     

Thermally induced vibration and stability of laminated composite plates with temperature-dependent properties

This paper presents a study on the buckling and vibration of initially stressed composite plates with temperature-dependent material properties in thermal environments. The initial stress is taken to be a combination of a pure bending stress and an axial stress. The temperature distribution in the plate is assumed to be uniform and linear in the transverse direction. The governing equations including the transverse shear deformation effects are established using the variational method. The effects of various parameters on the buckling and vibration behaviors of laminated plates with respective temperature-dependent and temperature-independent material properties are investigated. The buckling load and natural frequency are sensitive to the thermal stresses and initial stresses. Numerical results reveal that temperature-dependent material properties should be considered in the buckling and vibration analysis for laminated plates under thermal conditions.     

Thermoelastic buckling behavior of thick functionally graded rectangular plates

Thermoelastic buckling behavior of thick rectangular plate made of functionally graded materials is investigated in this article. The material properties of the plate are assumed to vary continuously through the thickness of the plate according to a power-law distribution. Three types of thermal loading as uniform temperature raise, nonlinear and linear temperature distribution through the thickness of plate are considered. The coupled governing stability equations are derived based on the Reddy’s higher-order shear deformation plate theory using the energy method. The resulted stability equations are decoupled and solved analytically for the functionally graded rectangular plates with two opposite edges simply supported subjected to different types of thermal loading. A comparison of the present results with those available in the literature is carried out to establish the accuracy of the presented analytical method. The influences of power of functionally graded material, plate thickness, aspect ratio, thermal loading conditions and boundary conditions on the critical buckling temperature of aluminum/alumina functionally graded rectangular plates are investigated and discussed in detail. The critical buckling temperatures of thick functionally graded rectangular plates with various boundary conditions are reported for the first time and can be served as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.     

基于Layerwise离散层理论的复合材料夹层板屈曲分析

复合材料夹层结构由于面板和芯层力学特性差异较大,屈曲分析时要分层考虑各层的剪切变形。基于Reddy的Layerwise离散层理论,假设每一层变形服从一阶剪切变形理论,在统一的位移场描述下,推导建立了一种用于复合材料夹层结构屈曲分析的四节点四边形板单元,并采用混合插值方法对单元的剪切锁定进行了修正。分别对三种典型的夹层板结构进行线性屈曲有限元分析,并将计算结果与文献中已有结果进行了对比。结果表明：本文的分析方法能离散考虑各层的力学特性,将结构离散为多层时,计算结果与三维弹性理论或高阶板理论吻合;将结构等效为单层时,计算结果与基于一阶剪切变形理论的文献结构吻合,验证了单元的有效性。     

Thermal buckling and post-buckling response of imperfect temperature-dependent sandwich FGM plates resting on elastic foundation

Post-buckling behaviour of sandwich plates with functionally graded material (FGM) face sheets under uniform temperature rise loading is considered. It is assumed that the plate is in contact with a Pasternak-type elastic foundation during deformation, which acts in both compression and tension. The derivation of equations is based on the first-order shear deformation plate theory. Thermomechanical non-homogeneous properties of FGM layers vary smoothly by the distribution of power law across the thickness, and temperature dependency of material constituents is taken into account. Using the non-linear von-Karman strain-displacement relations, the equilibrium and compatibility equations of imperfect sandwich plates with FGM face sheets are derived. The boundary conditions for the plate are assumed to be simply supported in all edges. The governing equations are reduced to two coupled equation in terms of stress function and lateral deflection. Employing the single mode approach combined with Galerkin technique, an approximate closed-form solution is presented to calculate the critical buckling temperature and post-buckling equilibrium path of the plate. Presented numerical examples contain the influences of power law index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation coefficients.     

求解矩形夹层板在集中载荷下的稳定问题

在考虑了横向切应力和横向正应力对夹层板稳定影响的情况下,给出了矩形夹层板结构屈曲失稳的控制方程、基本解以及边界条件。应用功的互等定理求解了在均布载荷作用下的矩形夹层板的屈曲失稳问题。     

Design optimization of truss-cored sandwiches with homogenization

Lightweight metallic sandwich plates comprising periodic truss cores and solid facesheets are optimally designed against minimum weights. Constitutive models of the truss core are developed using homogenization techniques which, together with effective single-layer sandwich approaches, form the basis of a two-dimensional (2D) single-layer sandwich model. The 2D model is employed to simulate the mechanical behaviors of truss-cored sandwich panels having a variety of core topologies. The types of loading considered include bending, transverse shear and in-plane compression. The validities of the 2D model predictions are checked against direct FE simulations on three-dimensional (3D) truss core sandwich structures. Optimizations using the 2D sandwich model are subsequently performed to determine the minimum weights of truss-cored sandwiches subjected to various failure constraints: overall and local buckling, yielding and facesheet wrinkling. The performances of the optimized truss core sandwiches with 4-rod unit cell and solid truss members and pyramidal unit cell with hollow truss members are compared with benchmark lightweight structures such as honeycomb-cored sandwiches, tetrahedral core sandwiches and hat-stiffened single layer plates.     

A comprehensive analysis of functionally graded sandwich plates: Part 2—Buckling and free vibration

The sinusoidal shear deformation plate theory, presented in the first part of this paper, is used to study the buckling and free vibration of the simply supported functionally graded sandwich plate. Effects of rotatory inertia are considered. The critical buckling load and the vibration natural frequency are investigated. Some available results for sandwich plates non-symmetric about the mid-plane can be retrieved from the present analysis. The influences of the transverse shear deformation, plate aspect ratio, side-to-thickness ratio and volume fraction distributions are studied. In addition, the effect of the core thickness, relative to the total thickness of the plate, on the critical buckling load and the eigenfrequencies is investigated.     

Vibration and buckling of laminated sandwich plates having interfacial imperfections

The vibration and buckling characteristics of sandwich plates having laminated stiff layers are studied for different degrees of imperfections at the layer interfaces using a refined plate theory. With this plate theory, the through thickness variation of transverse shear stresses is represented by piece-wise parabolic functions where the continuity of these stresses is satisfied at the layer interfaces by taking jumps in the transverse shear strains at the interfaces. The transverse shear stresses free condition at the plate top and bottom surfaces is also satisfied. The inter-laminar imperfections are represented by in-plane displacement jumps at the layer interfaces and characterized by a linear spring layer model. It is quite interesting to note that this plate model having all these refined features requires unknowns only at the reference plane. To have generality in the analysis, finite element technique is adopted and it is carried out with a new triangular element developed for this purpose, as any existing element cannot model this plate model. As there is no published result on imperfect sandwich plates, the problems of perfect sandwich plates and imperfect ordinary laminates are used for validation.     

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.     

Mechanical buckling and free vibration of thick functionally graded plates resting on elastic foundation using the higher order B-spline finite strip method

In this study, the mechanical buckling and free vibration of thick rectangular plates made of functionally graded materials (FGMs) resting on elastic foundation subjected to in-plane loading is considered. The third order shear deformation theory (TSDT) is employed to derive the governing equations. It is assumed that the material properties of FGM plates vary smoothly by distribution of power law across the plate thickness. The elastic foundation is modeled by the Winkler and two-parameter Pasternak type of elastic foundation. Based on the spline finite strip method, the fundamental equations for functionally graded plates are obtained by discretizing the plate into some finite strips. The results are achieved by the minimization of the total potential energy and solving the corresponding eigenvalue problem. The governing equations are solved for FGM plates buckling analysis and free vibration, separately. In addition, numerical results for FGM plates with different boundary conditions have been verified by comparing to the analytical solutions in the literature. Furthermore, the effects of different values of the foundation stiffness parameters on the response of the FGM plates are determined and discussed.     

Buckling of a sandwich symmetrical circular plate with varying mechanical properties of the core

This paper is devoted to analytical and numerical studies of global buckling of a sandwich circular plate. The mechanical properties of the plate core vary along its thickness, remaining constant in the facings. The middle surface of the plate is its symmetrical plane. The mathematical model of the plate is presented. The field of displacements is formulated using the proposed nonlinear hypothesis that generalizes the classical hypotheses. The equations of equilibrium are formulated based on the principle of stationary total potential energy. The proposed mathematical model of the displacements considers the shear effect. The numerical model of the plate is also formulated with a view to verify the analytical one. Numerical calculations are carried out for the chosen family of plates. The values of the critical load obtained by the analytical and numerical methods are compared. The effects of the material properties of the core and the change of the plate radius on the critical load intensity are presented.     

基于新修正偶应力理论的Mindlin层合板热稳定性分析

以新修正偶应力理论为基础,首次提出了机械载荷与热载荷共同作用下的微尺度Mindlin层合板热稳定性模型,该模型只引入一个材料尺度参数,通过虚功原理推导出了控制方程和边界条件,以四边简支方板为例,进行了热稳定性分析,应用纳维叶解法得到解析解。结果表明,所建模型可以捕捉到尺度效应。材料尺度参数值越大,屈曲临界温度越高;当跨厚比增大时,屈曲临界温度下降;随着板几何参数的增大,模型将退化为宏观模型;温度变化量越大,考虑热载荷作用下的屈曲临界载荷越大,尺度效应体现越显著。     

Thermal post-buckling analysis of imperfect laminated plates using a higher-order shear deformation theory

Thermal post-buckling analysis is presented for a simply supported, composite laminated plate subjected to uniform or non-uniform tent-like temperature loading. The initial geometrical imperfection of the plate is taken into account. The formulations are based on the Reddy's higher-order shear deformation plate theory, and include thermal effects. The analysis uses a mixed Galerkin-perturbation technique to determine thermal buckling loads and post-buckling equilibrium paths. Numerical examples cover the performances of perfect and imperfect, antisymmetrically angle-ply and symmetrically cross-ply laminated plates. The effects played by transverse shear deformation, thermal load ratio, plate aspect ratio, total number of plies, fiber orientation and initial geometrical imperfections are studied. Typical results are presented in dimensionless graphical form.     

Nonlinear stability of advanced sandwich cylindrical shells comprising porous functionally graded material and carbon nanotube reinforced composite layers under elevated temperature

The nonlinear stability of sandwich cylindrical shells comprising porous functionally graded material(FGM) and carbon nanotube reinforced composite(CNTRC)layers subjected to uniform temperature rise is investigated. Two sandwich models corresponding to CNTRC and FGM face sheets are proposed. Carbon nanotubes(CNTs) in the CNTRC layer are embedded into a matrix according to functionally graded distributions. The effects of porosity in the FGM and the temperature dependence of properties of all constituent materials are considered. The effective properties of the porous FGM and CNTRC are determined by using the modified and extended versions of a linear mixture rule, respectively. The basic equations governing the stability problem of thin sandwich cylindrical shells are established within the framework of the Donnell shell theory including the von K'arm'an-Donnell nonlinearity. These equations are solved by using the multi-term analytical solutions and the Galerkin method for simply supported shells.The critical buckling temperatures and postbuckling paths are determined through an iteration procedure. The study reveals that the sandwich shell model with a CNTRC core layer and relatively thin porous FGM face sheets can have the best capacity of thermal load carrying. In addition, unlike the cases of mechanical loads, porosities have beneficial effects on the nonlinear stability of sandwich shells under the thermal load. It is suggested that an appropriate combination of advantages of FGM and CNTRC can result in optimal efficiency for advanced sandwich structures.     

Kármán-type equations for a higher-order shear deformation plate theory and its use in the thermal postbuckling analysis

Kármán-type nonlinear large deflection equations are derived occnrding to the Reddy’s higher-order shear deformation plate theory and used in the thermal postbuckling analysis The effects of initial geometric imperfections of the plate areincluded in the present study which also includes th thermal effects.Simply supported,symmetric cross-ply laminated plates subjected to uniform or nomuniform parabolictemperature distribution are considered. The analysis uses a mixed GalerkinGolerkinperlurbation technique to determine thermal buckling louds and postbucklingequilibrium paths.The effects played by transverse shear deformation plate aspeclraio, total number of plies thermal load ratio and initial geometric imperfections arealso studied.     

剪切变形对FGM圆板轴对称屈曲的影响

基于一阶剪切变形板理论，推导了功能梯度材料圆形板在边界面内均布压力作用下的轴对称屈曲方程。在推导过程中，忽略了前屈曲耦合变形。利用一阶板理论与经典板理论屈曲方程之间在数学形式上的相似性，得到了一阶板理论下功能梯度材料圆板与经典板理论下均匀圆板临界屈曲载荷之间的解析关系。利用这个解析关系，可以直接从已有的较为简单的经典理论的结果，获得一阶板理论下功能梯度材料板的临界屈曲载荷。