Exact solution for functionally graded and layered magneto-electro-elastic plates

In this paper, an exact solution is presented for the multilayered rectangular plate made of functionally graded, anisotropic, and linear magneto-electro-elastic materials. While the edges of the plate are under simply supported conditions, general mechanical, electric and magnetic boundary conditions can be applied on both the top and bottom surfaces of the plate. The functionally graded material is assumed to be exponential in the thickness direction and the homogeneous solution in each layer is obtained based on the pseudo-Stroh formalism. For multilayered plate structure, the propagator matrix method is employed so that only a 5 × 5 system of linear algebraic equations needs to be solved. The exact solution is then applied to two functionally graded (exponential) sandwich plates made of piezoelectric BaTiO₃ and magnetostrictive CoFe₂O₄, under mechanical and electric loads on the top surface. While the numerical results clearly show the influence of the exponential factor, magneto-electro-elastic properties, and loading types on induced magneto-electric-elastic fields, they can also serve as benchmarks to numerical methods such as the finite and boundary element methods.     

Finite element analysis for geometrically nonlinear deformations of smart functionally graded plates using vertically reinforced 1-3 piezoelectric composite

In this paper, a finite element model has been developed for the geometrically nonlinear static analysis of simply supported functionally graded (FG) plates integrated with a patch of vertically reinforced 1-3 piezoelectric composite material acting as a distributed actuator. The material properties of the functionally graded substrate plate are assumed to be graded only in the thickness direction according to the power-law distribution in terms of the volume fractions of the constituents. The analysis of the electro-elastic coupled problem includes the transverse deformations of the overall plate to utilize the transverse normal actuation by the distributed actuator for counteracting the nonlinear deformations of smart functionally graded plates. The nonlinear governing equations of equilibrium are solved by using direct iteration method with under-relaxation. The numerical illustrations suggest the potential use of the distributed actuator made of vertically reinforced 1-3 piezoelectric composite material for active control of nonlinear deformations of smart functionally graded plates. The effect of variation of piezoelectric fiber orientation in the distributed actuator on its control authority for counteracting the nonlinear deformations of smart functionally graded plates has also been investigated.     

Geometrically nonlinear static and free vibration analysis of functionally graded piezoelectric plates

In this paper, nonlinear static and free vibration analysis of functionally graded piezoelectric plates has been carried out using finite element method under different sets of mechanical and electrical loadings. The plate with functionally graded piezoelectric material (FGPM) is assumed to be graded through the thickness by a simple power law distribution in terms of the volume fractions of the constituents. Only the geometrical nonlinearity has been taken into account and electric potential is assumed to be quadratic across the FGPM plate thickness. The governing equations are obtained using potential energy and Hamilton’s principle that includes elastic and piezoelectric effects. The finite element model is derived based on constitutive equation of piezoelectric material accounting for coupling between elasticity and electric effect using higher order plate elements. The present finite element is modeled with displacement components and electric potential as nodal degrees of freedom. Results are presented for two constituent FGPM plate under different mechanical boundary conditions. Numerical results for PZT-4/PZT-5H plate are given in dimensionless graphical forms. Effects of material composition and boundary conditions on nonlinear response are also studied. The numerical results obtained by the present model are in good agreement with the available solutions reported in the literature.     

Dynamic response of functionally graded circular cylindrical shells subjected to radial impulse load

Dynamic response of simply supported circular cylindrical shell made of functionally graded material (FGM) subjected to lateral impulse load is investigated. The effective material properties are assumed to vary continuously along the thickness direction according to a volume fraction power law distribution. First order shear deformation theory and Love’s first approximation theory are utilized in the equilibrium equations. Finally time response of displacement components is derived using mode superposition method. The influence of material composition, FGM configuration and geometrical parameters (length-to-radius and thickness-to-radius ratios) on the dynamic response is investigated. The results show that even though the shell is thin, the value of power law exponent has a considerable effect on the natural frequencies as well as the dynamic response of the functionally graded shell. Verification of the results of natural frequencies and time response of the FGM shell is achieved by making comparison with those available in the literature and those obtained using commercial software.     

An exact Peano Series solution for bending analysis of imperfect layered functionally graded neutral magneto-electro-elastic plates resting on elastic foundations

In this article, a three-dimensional solution is presented for the bending analysis of functionally graded and layered neutral magneto-electro-elastic plates resting on two-parameter elastic foundations, considering imperfect interfacial bonding. The equations of motion, Gauss's equations for electrostatics and magnetostatics, and boundary and interface conditions are satisfied exactly regardless of the number of layers. No assumptions on deformations, stresses, and magnetic and electric fields along the thickness direction are introduced. The interfacial imperfection is modeled using a generalized spring layer. The state-space method is employed for solving the governing partial differential equations. Effects of a two-parameter elastic foundation, gradient index, bonding imperfection, and applied mechanical and electrical loads on the response of the functionally graded magneto-electro-elastic plate are discussed. The obtained exact solution can serve as a benchmark for assessing the accuracy of layered functionally graded magneto-electro-elastic plate theories.     

功能梯度材料Timoshenko型剪切梁的自由振动分析

基于Timoshenko梁理论,研究各向异性功能梯度材料梁的自由振动。假设材料参数沿梁厚度方向按同一函数规律变化,建立了功能梯度材料梁的振动方程,求得简支条件下其自振频率表达式。通过算例,给出指数函数梯度变化Timoshenko梁的自振频率和模态图,结果表明不同梯度变化对材料结构动力响应有较大影响。该方法为发展功能梯度材料梁的设计与数值计算提供了理论依据。     

Natural frequencies of cracked functionally graded material plates by the extended finite element method

In this paper, the linear free flexural vibration of cracked functionally graded material plates is studied using the extended finite element method. A 4-noded quadrilateral plate bending element based on field and edge consistency requirement with 20 degrees of freedom per element is used for this study. The natural frequencies and mode shapes of simply supported and clamped square and rectangular plates are computed as a function of gradient index, crack length, crack orientation and crack location. The effect of thickness and influence of multiple cracks is also studied.     

压电元件驱动的功能梯度弹性薄板的屈曲

考虑功能梯度薄板,其上下表面嵌有压电执行元件.根据逆压电效应将电压转换成作用于板上的等效电载荷.假设梯度材料的弹性参数为板厚度方向坐标的幂函数,基于经典板理论,导出了功能梯度弹性薄板小挠度屈曲平衡微分方程.利用双三角级数展开法,得到了四边简支具有压电元件的功能梯度矩形板的临界屈曲载荷,在此基础上通过数值例子讨论了弹性板的几何尺寸、材料梯度指数的变化对临界电压(载荷)的影响.研究结果表明,材料的梯度指数对临界电压有重要影响,并且通过调整作用于执行元件上的电压的大小和方向,可实现对结构稳定性的有效控制.     

Static solutions for functionally graded simply supported plates

In this article mixed semi-analytical and analytical solutions are presented for a rectangular plate made of functionally graded (FG) material. All edges of a plate are under simply supported (diaphragm) end conditions and general stress boundary conditions can be applied on both top and bottom surface of a plate during solution. A mixed semi-analytical model consists in defining a two-point boundary value problem governed by a set of first-order ordinary differential equations in the plate thickness direction. Analytical solutions based on shear-normal deformation theories are also established to show the accuracy, simplicity and effectiveness of mixed semi-analytical model. The FG material is assumed to be exponential in the thickness direction and Poisson’s ratio is assumed to be constant.     

Elasticity solutions for a uniformly loaded rectangular plate of functionally graded materials with two opposite edges simply supported

While the expansion formula for displacements and the hypothesis that the material parameters can vary along the thickness direction in an arbitrary fashion are retained, the plate theory of functionally graded materials suggested by Mian and Spencer is extended in this paper in two aspects. First, the material is assumed to be transversely isotropic, rather than isotropic. Second, the tractions-free conditions on the top and bottom surfaces are replaced by the conditions of uniform loads applied on the surfaces. The governing equations are derived rigorously, which are expressed in terms of three components of the mid-plane displacement. The mid-plane deflection satisfies a fourth-order partial differential equation, which is similar to that in the classical plate theory (CPT). The two in-plane displacements satisfy two second-order partial differential equations and are coupled with the mid-plane deflection. Boundary conditions are expressed in the form of those adopted in the CPT. Three-dimensional elasticity solutions are presented for a uniformly loaded rectangular plate with two opposite edges simply supported and six different types of supports at the other two edges. Numerical results are given for a simply supported functionally graded plate, and good agreement with exact solutions is obtained.     

四边简支功能梯度矩形板的热屈曲分析

基于经典板理论,假设材料性质为板厚度方向坐标的幂函数,推导了功能梯度材料矩形板在热荷载作用下的平衡方程和稳定方程。给出了四边简支的功能梯度板在均匀受热时临界屈曲温度变化的封闭解,讨论了板的几何外形尺寸、相对厚度、梯度指数以及中面变形等因素对临界屈曲温度变化的影响。     

Levy Solution for Buckling Analysis of Functionally Graded Rectangular Plates

In this article, an analytical method for buckling analysis of thin functionally graded (FG) rectangular plates is presented. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. Based on the classical plate theory (Kirchhoff theory), the governing equations are obtained for functionally graded rectangular plates using the principle of minimum total potential energy. The resulting equations are decoupled and solved for rectangular plate with different loading conditions. It is assumed that the plate is simply supported along two opposite edges and has arbitrary boundary conditions along the other edges. The critical buckling loads are presented for a rectangular plate with different boundary conditions, various powers of FGM and some aspect ratios.     

Analysis of thick functionally graded plates by using higher-order shear and normal deformable plate theory and MLPG method with radial basis functions

Infinitesimal deformations of a functionally graded thick elastic plate are analyzed by using a meshless local Petrov–Galerkin (MLPG) method, and a higher-order shear and normal deformable plate theory (HOSNDPT). Two types of Radial basis functions RBFs, i.e. Multiquadrics and Thin Plate Splines, are employed for constructing the trial solutions, while a fourth-order Spline function is used as the weight/test function over a local subdomain. Effective material moduli of the plate, made of two isotropic constituents with volume contents varying only in the thickness direction, are computed using the Mori–Tanaka homogenization technique. Computed results for a simply supported aluminum/ceramic plate are found to agree well with those obtained analytically. Results for a plate with two opposite edges free and the other two simply supported agree very well with those obtained by analyzing three-dimensional deformations of the plate by the finite element method. The distributions of the deflection and stresses through the plate thickness are also presented for different boundary conditions. It is found that both types of basis functions give accurate values of plate deflection, but the multiquadrics give better values of stresses than the thin plate splines.     

Semi-analytical mechanical and thermal buckling analyses of 2D-FGM circular plates based on the FSDT

In the present paper, mechanical and thermal buckling analyses of two-directional functionally graded material (2D-FGM) circular plate are investigated. The motion equations have been derived based on the first-order shear deformation theory (FSDT) and power series method has been employed to solve the motion equations. Two different kinds of boundary condition including simply supported and fixed are considered. The material properties are assumed to vary in both transverse and radial directions according to power and exponential laws, respectively. Comparisons with available studies in the literature confirm the high accuracy of the current approach. The effects of geometrical parameters and 2D-FG power indices on the critical buckling load have been studied. It is shown that increase of modulus of elasticity of outer layers of plate due to higher presence of hard phase of FGM, in radius and thickness directions of the plate makes it possible to attain a more solid structure against mechanical buckling loads, while increase of coefficient of thermal expansion and coefficient of thermal conduction of outer layers of plate results in less stability against thermal buckling loads.     

功能梯度矩形板的强非线性共振分析

针对陶瓷-金属功能梯度矩形板,在给出非均匀材料应力-应变关系及非线性几何方程的基础上,应用虚功原理导出了横向简谐激励力作用下功能梯度板的非线性振动偏微分方程。对于四边简支约束功能梯度矩形板,通过位移函数的设定,利用伽辽金积分法推得了关于时间自变量的达芬型强非线性振动方程。针对强非线性系统的主共振问题,应用改进的多尺度法进行解析求解,得到了稳态运动下的幅频响应方程。通过数值算例,给出了功能梯度矩形板共振下的幅频曲线图和相图,讨论了激励幅值及频率等参数对系统非线性振动特性的影响,并对改进多尺度法和经典多尺度法的结果进行了比较。     

Three dimensional elasticity solution for static and dynamic analysis of multi-directional functionally graded thick sector plates with general boundary conditions

In this study, the three dimensional static and dynamic behavior of a thick sector plate made of two-directional functionally graded materials (2D-FGMs) is investigated. Material properties are assumed to be graded in both radial and thickness directions according to a simple power law distribution in terms of the volume fractions of the constituents. The governing equations are based on the 3D theory of elasticity. Employing 3D graded finite element method (GFEM) based on the Hamilton’s principle and Rayleigh–Ritz energy method, the equations are solved in space and time domains. In the case of static analysis, the sector plate is subjected to a uniform pressure load and for dynamic analysis is subjected to an impact loading. The effects of material gradient index, boundary condition and thickness to radius ratio of the sector plate on the static and dynamic responses are presented and discussed.     

3-D free vibration analysis of thick functionally graded annular sector plates on Pasternak elastic foundation via 2-D differential quadrature method

Early studies on annular sector plate vibrations were focused on two-dimensional theories, such as the classical plate theory and the first- and the higher-order shear deformation plate theories. These plate theories neglect transverse normal deformations and generally assume that a plane stress state of deformation prevails in the plate. These assumptions may be appropriate for thin plates. In this paper, free vibration of thick functionally graded annular sector plates with simply supported radial edges on a two-parameter elastic foundation, based on the three-dimensional theory of elasticity, using differential quadrature method for different circular edge conditions including simply supported-clamped, clamped–clamped, and free-clamped is investigated. A semi-analytical approach composed of differential quadrature method and series solution is adopted to solve the equations of motion. The material properties change continuously through the thickness of the plate, which can vary according to a power law, exponentially, or any other formulations in this direction. Some new results for the natural frequencies of the plate are prepared, which include the effects of elastic coefficients of foundation, boundary conditions, material and geometrical parameters. The new results can be used as benchmark solutions for future research.     

磁电弹功能梯度材料板中的波动特性

假设材料常数和电、 磁常数沿板厚方向呈梯度变化, 将电磁功能梯度材料板沿厚度方向划分为层单元, 建立单元的控制方程, 然后根据单元之间的连续条件将单元控制方程装配成系统的控制方程, 求解控制方程, 得到不同模态波的弥散曲线, 分析了电磁参数对波的弥散特性的影响, 同时研究了材料梯度对弹性位移、 静电势和静磁势分布的影响, 并对磁电功能梯度材料中波的六个特征波面进行了分析。计算结果表明材料梯度引起弹性位移、 静电势、 静磁势的分布集中于材料参数梯度减少的方向, 材料为正交各向异性弹性材料。      

Thermal buckling analysis of moderately thick functionally graded annular sector plates

In this article, thermal buckling analysis of moderately thick functionally graded annular sector plate is studied. The equilibrium and stability equations are derived using first order shear deformation plate theory. These equations are five highly coupled partial differential equations. By using an analytical method, the coupled stability equations are replaced by four decoupled equations. Solving the decoupled equations and satisfying the boundary conditions, the critical buckling temperature is found analytically. To this end, it is assumed that the annular sector plate is simply supported in radial edges and it has arbitrary boundary conditions along the circular edges. Thermal buckling of functionally graded annular sector plate for two types of thermal loading, uniform temperature rise and gradient through the thickness, are investigated. Finally, the effects of boundary conditions, power law index, plate thickness, annularity and sector angle on the critical buckling temperature of functionally graded annular sector plates are discussed in details.     

Three-dimensional elastodynamic solution for an arbitrary thick FGM rectangular plate resting on a two parameter viscoelastic foundation

A three-dimensional semi-analytic analysis based on the linear elasticity theory is offered to study the transient vibration characteristics of an arbitrarily thick, simply supported, functionally graded (FGM) rectangular plate, resting on a linear Winkler–Pasternak viscoelastic foundation, and subjected to general distributed driving forces of arbitrary temporal and spatial variations. The problem solution is obtained by adopting a laminate model in conjunction with the powerful state space solution technique involving a global transfer matrix and Durbin’s numerical Laplace inversion algorithm. Numerical calculations are carried out for the transient displacement and stress responses of aluminum-zirconia FGM square plates of selected thickness parameters and compositional gradients, resting on “soft” or “stiff” elastic foundations, under the action of moving transverse forces as well as uniformly distributed blast loads. Also, the response curves for the FGM plates are compared with those of equivalent bilaminate plates containing comparable total volume fractions of constituent materials. It is observed that the material gradient variation is substantially more influential on the dynamic stress concentrations induced across the plate thickness than on the displacement response of the inhomogeneous plates. In particular, the displacement response of the equivalent bilaminate plates can provide an accurate estimate for prediction of the dynamic response of the corresponding FGM plates, especially for thick plates resting on a stiff foundation. Limiting cases are considered and good agreements with the data available in the literature as well as with the computations made by using a commercial finite element package are obtained.