Elasticity solutions for functionally graded plates in cylindrical bending

The plate theory of functionally graded materials suggested by Mian and Spencer is extended to analyze the cylindrical bending problem of a functionally graded rectangular plate subject to uniform load. The expansion formula for displacements is adopted. While keeping the assumption that the material parameters can vary along the thickness direction in an arbitrary fashion, this paper considers orthotropic materials rather than isotropic materials. In addition, the traction-free condition on the top surface is replaced with the condition of uniform load applied on the top surface. The plate theory for the particular case Of cylindrical bending is presented by considering an infinite extent in the y-direction. Effects of boundary conditions and material inhomogeneity on the static response of functionally graded plates are investigated through a numerical example.     

Semi-analytical elasticity solutions for bi-directional functionally graded beams

Elasticity solutions are presented for bending and thermal deformations of functionally graded beams with various end conditions, using the state space-based differential quadrature method. The beams are assumed to be macroscopically isotropic, with Young’s modulus varying exponentially along the thickness and longitudinal directions, while Poisson’s ratio remaining constant. The state space method is adopted to obtain analytically the thickness variation of the elastic field and, when coupled with differential quadrature, the longitudinal discretization can be analyzed in an approximate manner. This approach is then validated by comparing the numerical results with the exact solutions for a special functionally graded beam and with finite element solutions. The influences of material gradient indices on the response of bi-directional functionally graded beams are finally investigated.     

Elasticity solutions for functionally graded annular plates subject to biharmonic loads

Based on England’s expansion formula for displacements, the elastic field in a transversely isotropic functionally graded annular plate subjected to biharmonic transverse forces on its top surface is investigated using the complex variables method. The material parameters are assumed to vary along the thickness direction in an arbitrary fashion. The problem is converted to determine the expressions of four analytic functions α (ζ), β (ζ), ? (ζ) and ψ (ζ) under certain boundary conditions. A series of simple and practical biharmonic loads are presented. The four analytic functions are constructed carefully in a biconnected annular region corresponding to the presented loads, which guarantee the single-valuedness of the mid-plane displacements of the plate. The unknown constants contained in the analytic functions can be determined from the boundary conditions that are similar to those in the plane elasticity as well as those in the classical plate theory. Numerical examples show that the material gradient index and boundary conditions have a significant influence on the elastic field.     

New assessment on the Saint-Venant solutions for functionally graded beams

A symplectic approach is proposed to investigate the Saint-Venant problem of functionally graded beams with Young's modulus varying exponentially in the axial direction and constant Poisson radio. A matrix state equation is derived with a shift-Hamiltonian operator matrix whose particular eigenvalues are proved to compose the basic solutions of the Saint-Venant problem. The present analyses demonstrate that the Saint-Venant solutions under simple extension and pure bending can be derived using either the direct expansion method or the rigid motion removing method.     

Analytical solutions for plane problem of functionally graded magnetoelectric cantilever beam

In this paper, an exact analytical solution is presented for a transversely isotropic functionally graded magneto-electro-elastic (FGMEE) cantilever beam, which is subjected to a uniform load on its upper surface, as well as the concentrated force and moment at the free end. This solution can be applied for any form of gradient distribution. For the basic equations of plane problem, all the partial differential equations governing the stress field, electric, and magnetic potentials are derived. Then, the expressions of Airy stress, electric, and magnetic potential functions are assumed as quadratic polynomials of the longitudinal coordinate. Based on all the boundary conditions, the exact expressions of the three functions can be determined. As numerical examples, the material parameters are set as exponential and linear distributions in the thickness direction. The effects of the material parameters on the mechanical, electric, and magnetic fields of the cantilever beam are analyzed in detail.     

Static analysis of anisotropic functionally graded magneto-electro-elastic beams subjected to arbitrary loading

This paper considers the analytical and semi-analytical solutions for anisotropic functionally graded magneto-electro-elastic beams subjected to an arbitrary load, which can be expanded in terms of sinusoidal series. For the generalized plane stress problem, the stress function, electric displacement function and magnetic induction function are assumed to consist of two parts, respectively. One is a product of a trigonometric function of the longitudinal coordinate (x) and an undetermined function of the thickness coordinate (z), and the other a linear polynomial of x with unknown coefficients depending on z. The governing equations satisfied by these z-dependent functions are derived. The analytical expressions of stresses, electric displacements, magnetic induction, axial force, bending moment, shear force, average electric displacement, average magnetic induction, displacements, electric potential and magnetic potential are then deduced, with integral constants determinable from the boundary conditions. The analytical solution is derived for beam with material coefficients varying exponentially along the thickness, while the semi-analytical solution is sought by making use of the sub-layer approximation for beam with an arbitrary variation of material parameters along the thickness. The present analysis is applicable to beams with various boundary conditions at the two ends. Two numerical examples are presented for validation of the theory and illustration of the effects of certain parameters.     

Homogenized and classical expressions for static bending solutions for functionally graded material Levinson beams

The exact relationship between the bending solutions of functionally graded material (FGM) beams based on the Levinson beam theory and those of the corresponding homogenous beams based on the classical beam theory is presented for the material properties of the FGM beams changing continuously in the thickness direction. The deflection, the rotational angle, the bending moment, and the shear force of FGM Levinson beams (FGMLBs) are given analytically in terms of the deflection of the reference homogenous Euler-Bernoulli beams (HEBBs) with the same loading, geometry, and end supports. Consequently, the solution of the bending of non-homogenous Levinson beams can be simplified to the calculation of transition coefficients, which can be easily determined by variation of the gradient of material properties and the geometry of beams. This is because the classical beam theory solutions of homogenous beams can be easily determined or are available in the textbook of material strength under a variety of boundary conditions. As examples, for different end constraints, particular solutions are given for the FGMLBs under specified loadings to illustrate validity of this approach. These analytical solutions can be used as benchmarks to check numerical results in the investigation of static bending of FGM beams based on higher-order shear deformation theories.     

Elasticity solutions for a transversely isotropic functionally graded circular plate subject to an axisymmetric transverse load qrk

This paper considers the bending of transversely isotropic circular plates with elastic compliance coefficients being arbitrary functions of the thickness coordinate, subject to a transverse load in the form of qr^k (k is zero or a finite even number). The differential equations satisfied by stress functions for the particular problem are derived. An elaborate analysis procedure is then presented to derive these stress functions, from which the analytical expressions for the axial force, bending moment and displacements are obtained through integration. The method is then applied to the problem of transversely isotropic functionally graded circular plate subject to a uniform load, illustrating the procedure to determine the integral constants from the boundary conditions. Analytical elasticity solutions are presented for simply-supported and clamped plates, and, when degenerated, they coincide with the available solutions for an isotropic homogenous plate. Two numerical examples are finally presented to show the effect of material inhomogeneity on the elastic field in FGM plates.     

Three-dimensional elastic solutions for functionally graded circular plates

Three-dimensional elastic solutions are obtained for a functionally graded thick circular plate subject to axisymmetric conditions. We consider a isotropic material where the Young modulus depends exponentially on the position along the thickness, while the Poisson ratio is constant. The solution method utilises a Plevako’s representation form which reduces the problem to the construction of a potential function satisfying a linear fourth-order partial differential equation. We write this potential function in terms of Bessel functions and we pointwise assign mixed boundary conditions. The analytic solution is obtained in a general form and explicitly presented by assuming transversal load on the upper face and zero displacements on the mantle; this is done by superposing the solutions of problems with suitably imposed radial displacement. We validate the solution by means of a finite element approach; in this way, we highlight the effects of the material inhomogeneity and the limits of the employed numerical method near the mantle, where the solution shows a large sensitivity to the boundary conditions.     

Microstructure-dependent couple stress theories of functionally graded beams

A microstructure-dependent nonlinear Euler-Bernoulli and Timoshenko beam theories which account for through-thickness power-law variation of a two-constituent material are developed using the principle of virtual displacements. The formulation is based on a modified couple stress theory, power-law variation of the material, and the von Kármán geometric nonlinearity. The model contains a material length scale parameter that can capture the size effect in a functionally graded material, unlike the classical Euler-Bernoulli and Timoshenko beam theories. The influence of the parameter on static bending, vibration and buckling is investigated. The theoretical developments presented herein also serve to develop finite element models and determine the effect of the geometric nonlinearity and microstructure-dependent constitutive relations on post-buckling response.     

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of the major threats to their service safety in engineering applications. This paper aims to clarify a long-standing controversy on the thermal instability type of simply-supported FGM beams. First, based on the Euler-Bernoulli beam theory and von Kármán geometric nonlinearity, a nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang's two-variable method is formulated. Second, an approximate analytic solution to the nonlinear integro-differential boundary value problem under a thermal-induced inhomogeneous force boundary condition is obtained by using a semiinverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finally, based on the free energy theorem, it is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermore, the nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.     

A meshless collocation method for the plane problems of functionally graded material beams and plates using the DRK interpolation

A meshless collocation method is developed for the static analysis of plane problems of functionally graded (FG) elastic beams and plates under transverse mechanical loads using the differential reproducing kernel (DRK) interpolation, in which the DRK interpolant is constructed by the randomly distributed nodes. A point collocation method based on this DRK interpolation is developed for the plane stress and strain problems of homogeneous and FG elastic beams and plates. It is shown that the present DRK interpolation-based collocation method is indeed a truly meshless approach with excellent accuracy and has a fast convergence rate.     

Fracture analysis of functionally graded coatings: plane deformation

A new multi-layered model for fracture analysis of functionally graded materials (FGMs) with arbitrarily varying elastic moduli under plane deformation has been developed. In this model, the FGM is divided into several sub-layers and in each sub-layer the shear modulus is assumed to be a linear function of the depth while the Poisson's ratio is assumed to be a constant. With this new model, an interface crack problem of a functionally graded coating bonded to a homogeneous half-plane under normal and shear loading is investigated. Employment of transfer matrix method and Fourier integral transform technique reduces the problem to a system of Cauchy singular integral equations which are solved numerically. Stress intensity factors of an interface crack are obtained for the cases of the shear modulus varying in an exponential manner and in a sinusoidal manner. Comparison of the present new model to other existing models shows that the new one is more efficient.     

考虑尺度影响的平面正交各向异性功能梯度微梁的屈曲分析

基于新修正偶应力理论,建立了能描述尺度效应的各向异性功能梯度微梁的屈曲分析模型。基于最小势能原理推导了控制方程及边界条件,并以简支梁为例分析了屈曲载荷及尺度效应受材料尺度参数和几何尺寸的影响。算例结果表明,在材料几何尺寸较小时,本文模型预测到的屈曲载荷明显大于传统理论的结果,有效地反映了尺度效应。几何尺寸较大时,尺度效应消失,本文模型将自动退化为传统宏观模型。模型反映出不同方向上的尺度参数对各向异性材料影响的效果不同。     

Basic solutions of multiple parallel symmetric mode-III cracks in functionally graded piezoelectric/piezomagnetic material plane

The Schmidt method is adopted to investigate the fracture problem of multiple parallel symmetric and permeable finite length mode-III cracks in a functionally graded piezoelectric/piezomagnetic material plane. This problem is formulated into dual integral equations, in which the unknown variables are the displacement jumps across the crack surfaces. In order to obtain the dual integral equations, the displacement jumps across the crack surfaces are directly expanded as a series of Jacobi polynomials. The results show that the stress, the electric displacement, and the magnetic flux intensity factors of cracks depend on the crack length, the functionally graded parameter, and the distance among the multiple parallel cracks. The crack shielding effect is also obviously presented in a functionally graded piezoelectric/piezomagnetic material plane with mul- tiple parallel symmetric mode-III cracks.     

Two-dimensional elasticity solution for bending of functionally graded beams with variable thickness

This paper studies the stress and displacement distributions of functionally graded beam with continuously varying thickness, which is simply supported at two ends. The Young’s modulus is graded through the thickness following the exponential-law and the Poisson’s ratio keeps constant. On the basis of two-dimensional elasticity theory, the general expressions for the displacements and stresses of the beam under static loads, which exactly satisfy the governing differential equations and the simply supported boundary conditions at two ends, are analytically derived out. The unknown coefficients in the solutions are approximately determined by using the Fourier sinusoidal series expansions to the boundary conditions on the upper and lower surfaces of the beams. The effect of Young’s modulus varying rules on the displacements and stresses of functionally graded beams is investigated in detail. The two-dimensional elasticity solution obtained can be used to assess the validity of various approximate solutions and numerical methods for the aforementioned functionally graded beams.     

A higher-order theory for static and dynamic analyses of functionally graded beams

The higher-order theory is extended to functionally graded beams (FGBs) with continuously varying material properties. For FGBs with shear deformation taken into account, a single governing equation for an auxiliary function F is derived from the basic equations of elasticity. It can be used to deal with forced and free vibrations as well as static behaviors of FGBs. A general solution is constructed, and all physical quantities including transverse deflection, longitudinal warping, bending moment, shear force, and internal stresses can be represented in terms of the derivatives of F. The static solution can be determined for different end conditions. Explicit expressions for cantilever, simply supported, and clamped-clamped FGBs for typical loading cases are given. A comparison of the present static solution with existing elasticity solutions indicates that the method is simple and efficient. Moreover, the gradient variation of Young’s modulus and Poisson’s ratio may be arbitrary functions of the thickness direction. Functionally graded Rayleigh and Euler–Bernoulli beams are two special cases when the shear modulus is sufficiently high. Moreover, the classical Levinson beam theory is recovered from the present theory when the material constants are unchanged. Numerical computations are performed for a functionally graded cantilever beam with a gradient index obeying power law and the results are displayed graphically to show the effects of the gradient index on the deflection and stress distribution, indicating that both stresses and deflection are sensitive to the gradient variation of material properties.     

The method of fundamental solutions for nonlinear functionally graded materials

In this paper, we investigate the application of the method of fundamental solutions (MFS) to two-dimensional steady-state heat conduction problems for both isotropic and anisotropic, single and composite (bi-materials), nonlinear functionally graded materials (FGMs). In the composite case, the interface continuity conditions are approximated in the same manner as the boundary conditions. The method is tested on several examples and its relative merits and disadvantages are discussed.     

Accurate analytical perturbation approach for large amplitude vibration of functionally graded beams

The present work derives the accurate analytical solutions for large amplitude vibration of thin functionally graded beams. In accordance with the Euler–Bernoulli beam theory and the von Kármán type geometric non-linearity, the second-order ordinary differential equation having odd and even non-linearities can be formulated through Hamilton's principle and Galerkin's procedure. This ordinary differential equation governs the non-linear vibration of functionally graded beams with different boundary constraints. Building on the original non-linear equation, two new non-linear equations with odd non-linearity are to be constructed. Employing a generalised Senator–Bapat perturbation technique as an ingenious tool, two newly formulated non-linear equations can be solved analytically. By selecting the appropriate piecewise approximate solutions from such two new non-linear equations, the analytical approximate solutions of the original non-linear problem are established. The present solutions are directly compared to the exact solutions and the available results in the open literature. Besides, some examples are selected to confirm the accuracy and correctness of the current approach. The effects of boundary conditions and vibration amplitudes on the non-linear frequencies are also discussed.