Semi-analytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations

Buckling analysis of the functionally graded viscoelastic circular plates has not been carried out so far. In the present paper, a series solution is developed for buckling analysis of radially graded FG viscoelastic circular plates with variable thickness resting on two-parameter elastic foundations, based on Mindlin's plate theory. The complex modulus approach in combination with the elastic-viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A comprehensive sensitivity analysis is carried out to evaluate effects of various parameters on the buckling load. Results reveal that the viscoelastic behavior of the materials may postpone the buckling occurrence and the stiffness reduction due to the section variations may be compensated by the graded material properties.     

梯度弹性基础上正交异性薄板的屈曲分析

基于双参数弹性基础模型,研究了梯度弹性基础上正交异性薄板的屈曲问题. 首先,基于能量法与变分原理,给出了梯度弹性基础上正交异性薄板的屈曲控制方程,并得到了梯度弹性基础刚度系数K₁ 与K₂的计算式;进而,通过将位移函数采用三角函数展开的方法,给出了单向压缩载荷作用下、四边简支正交异性弹性基础板屈曲载荷的计算式;在算例中,通过将该文的解退化到单纯的正交异性板,并与经典弹性解比较,证明了理论的正确性;最后,求解了弹性模量在厚度方向上呈幂律分布的梯度基础上的薄板屈曲问题,分析了基础上下表层材料弹性模量比与体积分数指数对屈曲载荷的影响.     

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.     

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.     

Creep buckling and post-buckling analysis of the laminated piezoelectric viscoelastic functionally graded plates

The creep buckling and post-buckling of the laminated piezoelectric viscoelastic functionally graded material (FGM) plates are studied in this research. Considering the transverse shear deformation and geometric nonlinearity, the Von Karman geometric relation of the laminated piezoelectric viscoelastic FGM plates with initial deflection is established. And then nonlinear creep governing equations of the laminated piezoelectric viscoelastic FGM plates subjected to an in-plane compressive load are derived on the basis of the elastic piezoelectric theory and Boltzmann superposition principle. Applying the finite difference method and the Newmark scheme, the whole problem is solved by the iterative method. In numerical examples, the effects of geometric nonlinearity, transverse shear deformation, the applied electric load, the volume fraction and the geometric parameters on the creep buckling and post-buckling of laminated piezoelectric viscoelastic FGM plates with initial deflection are investigated.     

INVESTIGATION OF THE THICKNESS VARIABILITY AND MATERIAL HETEROGENEITY EFFECTS ON FREE VIBRATION OF THE VISCOELASTIC CIRCULAR PLATES

The present study deals with free vibration analysis of variable thickness viscoelastic circular plates made of heterogeneous materials and resting on two-parameter elastic foundations in addition to their edge conditions.It is assumed that the viscoelastic material properties vary in the transverse and radial directions simultaneously.The complex modulus approach is employed in conjunction with the elastic-viscoelastic correspondence principle to obtain the solution.The governing equations are solved by means of a power series solution.Finally,a sensitivity analysis including evaluation of effects of various edge conditions,thickness variations,coefficients of the elastic foundation,and material loss factor and heterogeneity on the natural frequencies and modal loss factors is accomplished.     

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman's plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material(FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.     

Buckling analysis of functionally graded arbitrary straight-sided quadrilateral plates on elastic foundations

As a first endeavor, the buckling analysis of functionally graded (FG) arbitrary straight-sided quadrilateral plates rested on two-parameter elastic foundation under in-plane loads is presented. The formulation is based on the first order shear deformation theory (FSDT). The material properties are assumed to be graded in the thickness direction. The solution procedure is composed of transforming the governing equations from physical domain to computational domain and then discretization of the spatial derivatives by employing the differential quadrature method (DQM) as an efficient and accurate numerical tool. After studying the convergence of the method, its accuracy is demonstrated by comparing the obtained solutions with the existing results in literature for isotropic skew and FG rectangular plates. Then, the effects of thickness-to-length ratio, elastic foundation parameters, volume fraction index, geometrical shape and the boundary conditions on the critical buckling load parameter of the FG plates are studied.     

粘弹性大挠度圆板的轴对称弯曲

本文探讨粘弹性大挠度圆板的轴对称弯曲的基本方程和求解方法.用半逆解和摄动法分析挠度与膜力,对标准线性固体进行数例计算,并与小挠度理论相比较.全部方程与解答可退化得相应的弹性大挠度板的结果.     

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。     

A double-superposition global–local theory for vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates: a complex modulus approach

A higher-order global–local theory is proposed based on the double-superposition concept for free vibration and dynamic buckling analyses of viscoelastic composite/sandwich plates subjected to thermomechanical loads. In contrast to all theories proposed so far for analysis of the viscoelastic plates, the continuity conditions of the transverse shear and normal stresses at the layer interfaces and the nonzero traction conditions at the top and bottom surfaces of the sandwich plates are satisfied. Another novelty is that these conditions may be satisfied for viscoelastic plates with temperature-dependent material properties and nonlinear behaviors subjected to thermomechanical loads. Furthermore, transverse flexibility is also taken into account. Some dynamic buckling/wrinkling analyses of the viscoelastic plates are performed in the present paper, for the first time. Comparisons made between results of the paper and results reported by well-known references confirm the accuracy and the efficiency of the proposed theory and the relevant solution algorithm.     

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.     

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.     

An analytical solution for buckling of moderately thick functionally graded sector and annular sector plates

In this article, an analytical solution for buckling of moderately thick functionally graded (FG) sectorial plates is presented. It is assumed that the material properties of the FG plate vary through the thickness of the plate as a power function. The stability equations are derived according to the Mindlin plate theory. By introducing four new functions, the stability equations are decoupled. The decoupled stability equations are solved analytically for both sector and annular sector plates with two simply supported radial edges. Satisfying the edges conditions along the circular edges of the plate, an eigenvalue problem for finding the critical buckling load is obtained. Solving the eigenvalue problem, the numerical results for the critical buckling load and mode shapes are obtained for both sector and annular sector plates. Finally, the effects of boundary conditions, volume fraction, inner to outer radius ratio (annularity) and plate thickness are studied. The results for critical buckling load of functionally graded sectorial plates are reported for the first time and can be used as benchmark.     

Elastic buckling of rectangular plates under linearly varying in-plane normal load with a circular cutout

The elastic buckling behavior of rectangular perforated plates was studied by using the finite element method in this study. Circular cutout was chosen at different locations along the principal x-axis of plates subjected to linearly varying loading in order to evaluate the effect of cutout location on the buckling behavior of plates. The results show that the center of a circular hole should not be placed at the end half of the outer panel for all loading patterns. Furthermore, the presence of a circular hole always causes a decrease in the elastic buckling load of plates subjected to bending, even if the circular hole is not in the outer panel.     

功能梯度与均匀圆板静动态解之间的相似转换关系

基于经典板理论,研究了功能梯度材料圆板的轴对称弯曲、屈曲和自由振动解与相应的均匀材料圆板解之间的转换关系.通过消去拉-弯耦合项得到了以挠度函数表示的功能梯度圆板的弯曲、屈曲和自由振动控制方程.分析功能梯度圆板与均匀圆板的控制方程之间的相似性,得到了功能梯度材料圆板与均匀圆板的解之间解的相似转换关系,在假定FGM圆板的材料性质沿厚分别以幂函数和指数函数的度变规律后,给出了相应的转换系数的解析表达式.该系数集中反映了功能梯度圆板的材料非均匀性.在已知均匀材料圆板轴对称解的条件下,可将功能梯度材料圆板轴对称问题的求解转化为相似转换系数的计算问题.这一方法可为非均匀板的求解提供了十分便捷有效的途径,而且便于工程应用.     

Stability analysis of functionally graded rectangular plates under nonlinearly varying in-plane loading resting on elastic foundation

In this research work, an exact analytical solution for buckling of functionally graded rectangular plates subjected to non-uniformly distributed in-plane loading acting on two opposite simply supported edges is developed. It is assumed that the plate rests on two-parameter elastic foundation and its material properties vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the classical plate theory based on exact neutral surface position is employed to derive the governing stability equations. Considering Levy-type solution, the buckling equation reduces to an ordinary differential equation with variable coefficients. An exact analytical solution is obtained for this equation in the form of power series using the method of Frobenius. By considering sufficient terms in power series, the critical buckling load of functionally graded plate with different boundary conditions is determined. The accuracy of presented results is verified by appropriate convergence study, and the results are checked with those available in related literature. Furthermore, the effects of power of functionally graded material, aspect ratio, foundation stiffness coefficients and in-plane loading configuration together with different combinations of boundary conditions on the critical buckling load of functionally graded rectangular thin plate are studied.     

Buckling and large deflection behaviors of radially functionally graded ring-stiffened circular plates with various boundary conditions

The buckling and large deflection behaviors of axis-symmetric radially functionally graded (RFG) ring-stiffened circular plates are investigated by the dynamic relaxation (DR) method combined with the finite difference discretization technique. The material properties of the constituent components of the RFG plate are assumed to vary continuously according to the Mori-Tanaka distribution along the radial direction. The nonlinear governing equations are obtained in the incremental form based on the first-order shear deformation plate theory (FSDT) and the von Karman relations for large deflection. In the buckling analysis, an external in-plane load is applied to the plate incrementally so that, in each load-step, the incremental form of the governing equations can be solved by a numerical code prepared based on the DR method. After converging the DR code in the first increment, the latter load-step is added to the previous one, and the program is repeated again. The critical buckling load is determined from the compressive load-displacement curve obtained by solving the incremental form of the governing equa- tions. Based on the present incremental form of formulation, a bending analysis can also be conducted if the whole load is applied simultaneously. Finally, a detailed parametric study is carried out to investigate the influences of various boundary conditions, grading indices, thickness-to-radius ratios, stiffener’s positions and depths on the critical buckling load, and displacements and stresses resulted from the bending analysis. It is observed that the effect of the stiffener on the results is much greater in the functionally graded plate with higher material grading indices. The results also reveal that, by increasing the depth of the stiffer, the values of ascending the critical buckling load are approximately identical for both simply supported and clamped boundary conditions.     

On the buckling of the elastic and viscoelastic composite circular thick plate with a penny-shaped crack

Buckling problem of the elastic and viscoelastic rotationally symmetric thick circular plate with a penny-shaped crack is investigated. It is supposed that the crack edges have a small initial rotationally symmetric imperfection. The lateral boundary of the plate is clamped and the clamp compresses this plate circumferentially and inwards by a fixed radial displacement. The investigations are carried out in the framework of the exact geometrically non-linear equations of the theory of viscoelasticity and as a buckling criterion the case for which the initial imperfection of the crack edges start to increase indefinitely is taken. Numerical results are obtained using the Laplace transform and finite element method and are compared with the known ones for elastic composites.     

The validity range of CPT and Mindlin plate theory in comparison with 3-D vibrational analysis of circular plates on the elastic foundation

The validity and the range of applicability of the classical plate theory (CPT) and the first-order shear deformation plate theory, also called Mindlin plate theory (MPT), in comparison with three-dimensional (3-D) p-Ritz solution are presented for freely vibrating circular plates on the elastic foundation with different boundary conditions. In order to achieve this purpose, a study of the 3-D elasticity solution is carried out to determine the free vibration frequencies of clamped, simply supported and free circular plates resting on an elastic foundation. The Pasternak model with adding a shear layer to the Winkler model is used for describing the elastic foundation. In addition to being employed the p-Ritz algorithm, the analysis is based on the linear, small strain and 3-D elasticity theory. In this analysis method, a set of orthogonal polynomial series in a cylindrical polar coordinate system is used to arrive eigenvalue equation yielding the natural frequencies for the circular plates. The accuracy of these results is verified by appropriate convergence studies and checked with the available literature and the MPT. Furthermore, the effect of the foundation stiffness parameters, thickness-radius ratio, and different boundary conditions on the ill-conditioning of the mass matrix as well as on the vibration behavior of the circular plates is investigated. Afterwards, the validity and the range of applicability of the results obtained on the basis of the CPT and MPT for a thin and moderately thick circular plate with different values of the foundation stiffness parameters are graphically presented through comparing them with those obtained by the present 3-D p-Ritz solution. Finally, the phenomenon of mode shape switching is investigated in graphical forms for a wide range of the Winkler foundation stiffness parameters.