Imperfection sensitivity of the post-buckling behavior of higher-order shear deformable functionally graded plates

This paper investigates the sensitivity of the post-buckling behavior of shear deformable functionally graded plates to initial geometrical imperfections in general modes. A generic imperfection function that takes the form of the product of trigonometric and hyperbolic functions is used to model various possible initial geometrical imperfections such as sine type, local type, and global type imperfections. The formulations are based on Reddy’s higher-order shear deformation plate theory and von Karman-type geometric nonlinearity. A semi-analytical method that makes use of the one-dimensional differential quadrature method, the Galerkin technique, and an iteration process is used to obtain the post-buckling equilibrium paths of plates with various boundary conditions that are subjected to edge compressive loading together with a uniform temperature change. Special attention is given to the effects of imperfection parameters, which include half-wave number, amplitude, and location, on the post-buckling response of plates. Numerical results presented in graphical form for zirconia/aluminum (ZrO₂/Al) graded plates reveal that the post-buckling behavior is very sensitive to the L2-mode local type imperfection. The influences of the volume fraction index, edge compression, temperature change, boundary condition, side-to-thickness ratio and plate aspect ratio are also discussed.     

Analyzing the effects of jump phenomenon in nonlinear vibration of thin circular functionally graded plates

In this paper, the nonlinear vibration of a thin circular functionally graded material plates is studied. The plate thickness is constant, and the material properties of the plate are assumed to vary continuously through the thickness. The governing equations and boundary conditions are extracted. The assumed-time-mode method is used to analyze these equations. The time variable is eliminated by assuming a harmonic response for nonlinear vibration and using Kantorovich time averaging technique. Utilizing shooting and Runge–Kutta methods, the set of first-order nonlinear differential equations are solved. The effect of volume fraction index in free and forced vibration response and jump phenomenon is studied. The results show that jump phenomenon occur according to volume fraction index and uniform temperature in the special frequencies of forced vibration response.     

功能梯度石墨烯增强复合材料板的自由振动

基于高阶函数剪切变形板理论,研究了可移动简支边界条件下功能梯度石墨烯增强复合材料板的自由振动。利用修正的Halpin-Tsai模型估计了复合材料的等效杨氏模量。基于Hamilton原理构建其偏微分方程。通过Navier方法计算得到了板的量纲为一的频率,并与已有成果进行了比较,验证了本文计算模型的优越性。数值计算结果表明:功能梯度石墨烯增强复合材料板的量纲为一的基频受石墨烯小块的分布模式影响很大,随着石墨烯含量的增加而单调增加;固定石墨烯小块的厚度和长宽比,量纲为一的基频随着长厚比的增加而单调增加。     

Nonlinear vibration analysis of piezo-thermo-electrically actuated functionally graded circular plates

A theoretical model for geometrically nonlinear vibration analysis of thermo-piezoelectrically actuated circular plates made of functionally grade material (FGM) is presented based on Kirchhoff’s–Love hypothesis with von-Karman type geometrical large nonlinear deformations. The material properties of the FG core plate are assumed to be graded in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. Dynamic equations and boundary conditions including thermal, elastic and piezoelectric couplings are formulated and solutions are derived. An exact series expansion method combined with perturbation approach is used to model the nonlinear thermo-electro-mechanical vibration behavior of the structure. Control of the FG plate’s nonlinear deflections and natural frequencies using high control voltages is studied and their nonlinear effects are evaluated. Numerical results for FG plates with various mixtures of ceramic and metal are presented in dimensionless forms. A parametric study is also undertaken to highlight the effects of the thermal environment, applied actuator voltage and material composition of the FG core plate on the nonlinear vibration characteristics of the composite structure.     

An efficient shear deformation theory for vibration of functionally graded plates

This paper presents an efficient shear deformation theory for vibration of functionally graded plates. The theory accounts for parabolic distribution of the transverse shear strains and satisfies the zero traction boundary conditions on the surfaces of the plate without using shear correction factors. The mechanical properties of functionally graded plate are assumed to vary according to a power law distribution of the volume fraction of the constituents. Equations of motion are derived from the Hamilton??s principle. Analytical solutions of natural frequency are obtained for simply supported plates. The accuracy of the present solutions is verified by comparing the obtained results with those predicted by classical theory, first-order shear deformation theory, and higher-order shear deformation theory. It can be concluded that the present theory is not only accurate but also simple in predicting the natural frequencies of functionally graded plates.     

Mathematical solution for nonlinear cylindrical bending of sigmoid functionally graded plates

Problems of nonlinear cylindrical bending of sigmoid functionally graded plates in which material properties vary through the thickness are considered. The variation of the material properties follows two power-law distributions in terms of the volume fractions of constituents. The nonlinear strain-displacement relations in the von Kármán sense are used to study the effect of geometric nonlinearity. The governing equations are reduced to a linear differential equation with nonlinear boundary conditions, yielding a simple solution procedure. Numerical results are presented to show the effect of the material distribution on the deflections and stresses.     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates.Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton's principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-ordex theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.     

Free vibration of functionally graded sandwich plates using four-variable refined plate theory

This paper uses the four-variable refined plate theory (RPT) for the free vibration analysis of functionally graded material (FGM) sandwich rectangular plates. Unlike other theories, there are only four unknown functions involved, as compared to five in other shear deformation theories. The theory presented is variationally consistent and strongly similar to the classical plate theory in many aspects. It does not require the shear correction factor, and gives rise to the transverse shear stress variation so that the transverse shear stresses vary parabolically across the thickness to satisfy free surface conditions for the shear stress. Two common types of FGM sandwich plates are considered, namely, the sandwich with the FGM facesheet and the homogeneous core and the sandwich with the homogeneous facesheet and the FGM core. The equation of motion for the FGM sandwich plates is obtained based on Hamilton’s principle. The closed form solutions are obtained by using the Navier technique. The fundamental frequencies are found by solving the eigenvalue problems. The validity of the theory is shown by comparing the present results with those of the classical, the first-order, and the other higher-order theories. The proposed theory is accurate and simple in solving the free vibration behavior of the FGM sandwich plates.     

Low-velocity impact response of geometrically nonlinear functionally graded graphene platelet-reinforced nanocomposite plates

Nonlinear Dynamics - This paper investigates the low-velocity impact response of functionally graded multilayer nanocomposite plates reinforced with a low content of graphene nanoplatelets (GPLs)...     

弹性地基上含孔隙功能梯度材料板的自由振动和动力响应

为研究弹性地基上含孔隙的材料特性沿厚度呈Sigmoid函数变化的功能梯度材料(S-FGM)板的振动特性,本文基于改进的Voigt模型,分别建立了孔隙为均匀分布和非均匀分布两种类型的功能梯度材料的物性参数模型.根据复合材料薄板理论导出了弹性地基上含孔隙的功能梯度材料板的运动方程,用伽辽金法寻求四边简支边界条件下板自由振动...     

Three-dimensional free vibration analysis of functionally graded piezoelectric annular plates on elastic foundations

Three-dimensional free vibration analysis of functionally graded piezoelectric (FGPM) annular plates resting on Pasternak foundations with different boundary conditions is presented. The material properties are assumed to have an exponent-law variation along the thickness. A semi-analytical approach which makes use of state-space method in thickness direction and one-dimensional differential quadrature method in radial direction is utilized to obtain the influences of the Winkler and shearing layer elastic coefficients of the foundations on the non-dimensional natural frequencies of functionally graded piezoelectric annular plates. The analytical solution in the thickness direction can be acquired using the state-space method and approximate solution in the radial direction can be obtained using the one-dimensional differential quadrature method. Numerical results are given to demonstrate the convergency and accuracy of the present method. The influences of the material property graded index, circumferential wave number and thickness of the annular plate on the dynamic behavior are also investigated. Since three-dimensional free vibration analysis of FGPM annular plates on elastic foundations has not been implemented before, the new results can be used as benchmark solutions for future researches.     

Free vibration analysis of elastically supported functionally graded annular plates subjected to thermal environment

Free vibration analysis of functionally graded (FG) thin-to-moderately thick annular plates subjected to thermal environment and supported on two-parameter elastic foundation is investigated. The material properties are assumed to be temperature-dependent and graded in the thickness direction. The equations of motion and the related boundary conditions, which include the effects of initial thermal stresses, are derived using the Hamilton’s principle based on the first order shear deformation theory (FSDT). The initial thermal stresses are obtained by solving the thermoelastic equilibrium equations. Differential quadrature method (DQM) as an efficient and accurate numerical tool is adopted to solve the thermoelastic equilibrium equations and the equations of motion. The formulations are validated by comparing the results in the limit cases with the available solutions in the literature for isotropic and FG circular and annular plates. The effects of the temperature rise, elastic foundation coefficients, the material graded index and different geometrical parameters on the frequency parameters of the FG annular plates are investigated. The new results can be used as benchmark solutions for future researches.     

碳纳米管增强型复合材料功能梯度板的自由振动模型与尺度效应

基于一种新的各向异性修正偶应力理论,建立了碳纳米管增强复合材料功能梯度板的自由振动模型。该模型能够描述尺度效应,且仅包含一个尺度参数。基于一阶剪切变形理论和哈密顿原理推演了板的运动微分方程,并以四边简支板为例给出了自振频率的解析解。讨论了板的几何尺寸、碳纳米管体分比含量和分布方式等因素对板的自振频率的影响。结果表明,本文模型所预测的板的自振基频总是高于经典弹性理论的Mindlin板模型的预测结果,两者间的差异在板的几何尺寸接近尺度参数的值时非常明显,且会随着板的几何尺寸的增大而逐渐消失。     

Bending of functionally graded piezoelectric rectangular plates

By introducing two displacement functions as well as two stress functions, two independent state equations with variable coefficients are derived from the three-dimensional theory equations of piezoelasticity for transverse isotropy. A laminated approximation is used to transform the state equations to those with constant coefficients in each sub-layer. The bending problem of a functionally graded rectangular plate is then analyzed based on the state equations. Numerical results are presented and the effect of material gradient index is discussed. Supported by the National Natural Sciences Foundation of China (No. 10002016).     

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.     

Thermoviscoelastic deformations of functionally graded thin plates

By considering the influence of the mid-plane strains due to the inhomogeneous property of FGMs across the thickness, a constitutive equation of thermoviscoelastic functionally graded thin plates is reduced on the basis of the Kirchhoff's hypothesis for the classical plate theory. The corresponding simplified Gurtin's type variational principle of functionally graded thin plates is presented by means of the modern convolution bilinear forms. By using the Navier analytic method or combining the Ritz method in the spatial domain and the Legendre interpolation method in the temporal domain, the static thermoelastic deformations or quasi-static thermoviscoelastic deformations of functionally graded thin plates under mechanical or thermal loads are studied.     

弹性地基加肋功能梯度板自由振动分析的无网格法

基于一阶剪切变形理论和移动最小二乘近似研究Winkler弹性地基上加肋功能梯度板的固有频率。假设功能梯度板的材料性质沿厚度方向按幂函数连续变化,基于物理中面和移动最小二乘近似分别推导功能梯度板和肋条的动能和势能,再通过引入位移协调条件,建立板和肋条节点参数转换关系,叠加两者的总能量,然后利用Hamilton原理推导加肋功能梯度板自由振动控制方程。采用完全转换法施加边界条件。通过将本文的计算结果与有限元以及文献的结果对比,验证方法的收敛性以及准确性。     

Efficient higher-order shear deformation theories for bending and free vibration analyses of functionally graded plates

In this paper, various efficient higher-order shear deformation theories are presented for bending and free vibration analyses of functionally graded plates. The displacement fields of the present theories are chosen based on cubic, sinusoidal, hyperbolic, and exponential variations in the in-plane displacements through the thickness of the plate. By dividing the transverse displacement into the bending and shear parts and making further assumptions, the number of unknowns and equations of motion of the present theories is reduced and hence makes them simple to use. Equations of motion are derived from Hamilton’s principle. Analytical solutions for deflections, stresses, and frequencies are obtained for simply supported rectangular plates. The accuracy of the present theories is verified by comparing the obtained results with the exact three-dimensional (3D) and quasi-3D solutions and those predicted by higher-order shear deformation theories. Numerical results show that all present theories can archive accuracy comparable to the existing higher-order shear deformation theories that contain more number of unknowns.     

Nonlinear free vibration and post-buckling analysis of functionally graded beams on nonlinear elastic foundation

In this study, simple analytical expressions are presented for large amplitude free vibration and post-buckling analysis of functionally graded beams rest on nonlinear elastic foundation subjected to axial force. Euler–Bernoulli assumptions together with Von Karman’s strain–displacement relation are employed to derive the governing partial differential equation of motion. Furthermore, the elastic foundation contains shearing layer and cubic nonlinearity. He’s variational method is employed to obtain the approximate closed form solution of the nonlinear governing equation. Comparison between results of the present work and those available in literature shows the accuracy of this method. Some new results for the nonlinear natural frequencies and buckling load of the FG beams such as the effect of vibration amplitude, elastic coefficients of foundation, axial force, and material inhomogenity are presented for future references.     

Thermoelastic buckling response of thick functionally graded plates

The thermoelastic buckling behavior of a thick plate made of a functionally graded material is investigated in this paper by using an exponential shear deformation plate theory. A simple power law based on the rule of mixtures is used to estimate the effective material properties as functions of the plate thickness. The neutral surface position for such functionally graded plates is determined on the basis of the nonlinear strain-displacement relations. Uniform, linear, and nonlinear temperature distributions across the plate are considered. An analytical approach is presented to find the critical buckling temperature, which can be used in engineering calculations. A numerical solution of the problem with the use of an exponential dependence for shear strains is presented. The results obtained are compared with available data.