Axisymmetric bending of functionally graded circular magneto-electro-elastic plates

Axisymmetric bending of functionally graded circular magneto-electro-elastic plates of transversely isotropic materials is analyzed based on linear three-dimensional theory of elasticity coupled with magnetic and electric fields. The transverse loads are expanded in Fourier-Bessel series and therefore can be arbitrarily distributed along the radial direction. The radial distributions of the displacements are assumed in combination of Fourier-Bessel series and polynomials as well as the electric potential and magnetic potential. If the material properties obey the exponential law along the thickness of the plate, two three-dimensional exact solutions for two unusual boundary conditions can be derived since they satisfy the governing equations and specified boundary conditions point by point. For simply supported or clamped boundary, the obtained solutions satisfy the governing equations exactly and the boundary conditions approximately. A layer wise model is also introduced to treat with the plates whose material property components vary independently and arbitrarily along the thickness of the plates. The numerical results are finally tabulated and plotted to demonstrate the presented method and agree well with those from finite element methods.     

Circumferential wave in magneto-electro-elastic functionally graded cylindrical curved plates

Piezoelectric-piezomagnetic functionally graded materials (FGM), with a gradual change of the mechanical and electromagnetic properties, have greatly applying promises. Based on Legendre orthogonal polynomial series expansion approach, a dynamic solution is presented for the propagation of circumferential harmonic waves in piezoelectric-piezomagnetic FGM cylindrical curved plates. The materials properties are assumed to vary in the direction of the thickness according to a known variation law. The dispersion curves of the piezoelectric-piezomagnetic FGM cylindrical curved plate and the corresponding non-piezoelectric and non-piezomagnetic cylindrical curved plates are calculated to show the influences of the piezoelectricity and piezomagnetism. Electric potential and magnetic potential distributions are also obtained to illustrate the different influences of the piezoelectricity and piezomagnetism. Finally, a cylindrical curved plate at a different ratio of radius to thickness is calculated to show the influence of the ratio on the piezoelectric effect and piezomagnetic effect.     

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.     

Three-dimensional analysis of doubly curved functionally graded magneto-electro-elastic shells

Three-dimensional (3D) solutions for the static analysis of doubly curved functionally graded (FG) magneto-electro-elastic shells are presented by an asymptotic approach. In the present formulation, the twenty-nine basic equations are firstly reduced to ten differential equations in terms of ten primary variables of elastic, electric and magnetic fields. After performing through the mathematical manipulation of nondimensionalization, asymptotic expansion and successive integration, we finally obtain recurrent sets of two-dimensional (2D) governing equations for various order problems. These 2D governing equations are merely those derived in the classical shell theory (CST) based on the extended Love–Kirchhoffs' assumptions. Hence, the CST-type governing equations are derived as a first-order approximation to the 3D magneto-electro-elasticity. The leading-order solutions and higher-order corrections can be determined by treating the CST-type governing equations in a systematic and consistent way. The 3D solutions for the static analysis of doubly curved multilayered and FG magneto-electro-elastic shells are presented to demonstrate the performance of the present asymptotic formulation. The coupling magneto-electro-elastic effect on the structural behavior of the shells is studied.     

Frictionless contact of a functionally graded magneto-electro-elastic layered half-plane under a conducting punch

This paper investigates the two-dimensional frictionless contact problem of a functionally graded magneto-electro-elastic materials (FGMEEMs) layered half-plane under a rigid flat or a cylindrical punch. It is assumed that the punch is a perfect electro-magnetic conductor with a constant electric potential and a constant magnetic potential. The magneto-electro-elastic (MEE) properties of the FGMEEM layer vary exponentially along the thickness direction. Using the Fourier transform technique, the contact problem can be reduced to Cauchy singular integral equations, which are then solved numerically to determine the normal contact stress, electric displacement and magnetic induction on the contact surface. Numerical results show that the gradient index, punch geometry and magneto-electro-mechanical loads have a significant effect on the contact behavior of FGMEEMs.     

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.     

Finite element analysis of smart functionally graded plates

This paper deals with the derivation of a finite element model for the static analysis of functionally graded (FG) plates integrated with a layer of piezoelectric fiber reinforced composite (PFRC) material. The layer of PFRC material acts as the distributed actuator of the FG plates. The Young’s modulus of the FG plate is assumed to vary exponentially along the thickness of the plate while the Poisson’s ratio is assumed to be constant over the domain of the plate. The finite element model has been verified with the exact solutions for both thick and thin plates. Emphasis has been placed on investigating the effect of variation of piezoelectric fiber angle in the PFRC layer on its actuating capability of the FG plates. The finite element solutions also revealed that the activated PFRC layer is more effective in controlling the deformations of the FG plates when the layer is attached to the surface of the FG plate with minimum stiffness than when it is attached to the surface of the same with maximum stiffness.     

An exact solution for transversely isotropic,simply supported and layered rectangular plates

Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.     

Three-dimensional elasticity solutions for bending of generally supported thick functionally graded plates

Three-dimensional elasticity solutions for static bending of thick functionally graded plates are presented using a hybrid semi-analytical approach-the state-space based differential quadrature method （SSDQM）. The plate is generally supported at four edges for which the two-way differential quadrature method is used to solve the in-plane variations of the stress and displacement fields numerically. An approximate laminate model （ALM） is exploited to reduce the inhomogeneous plate into a multi-layered laminate, thus applying the state space method to solve analytically in the thickness direction. Both the convergence properties of SSDQM and ALM are examined. The SSDQM is validated by comparing the numerical results with the exact solutions reported in the literature. As an example, the Mori-Tanaka model is used to predict the effective bulk and shear moduli. Effects of gradient index and aspect ratios on the bending behavior of functionally graded thick plates are investigated.     

Wave propagation analysis of a functionally graded magneto-electro-elastic nanobeam rest on Visco-Pasternak foundation

Wave propagation analysis of a nanobeam made of functionally graded magneto-electro-elastic materials with rectangular cross section rest on Visco-Pasternak foundation is studied in this paper. For modeling the axial, rotation and transverse deformations, Timoshenko beam model is used. Fundamental magneto-electro-elastic equations of the model are derived for a general functionally graded beam excited to electric and magnetic potentials. Surface elasticity is employed for more confident modeling the behavior of nanobeam. Using Hamilton principle and calculation of kinetic and strain energies, the equations of motion are derived. Considering the harmonic wave propagation of infinite domain yields characteristic equation of the system in terms of different parameters of model. The effects of different parameters such as non-homogeneous index, wave number and residual surface stress are investigated on the different phase velocities corresponding to modes of deformation. One can find that increasing the non-homogeneous index and wave number leads to decrease in wave propagation phase velocities.     

Dynamic internal crack problem of a functionally graded magneto-electro-elastic strip

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic strip containing an internal crack perpendicular to the boundary is investigated. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Numerical results show the effects of loading combination parameter, material gradient parameter and crack configuration on the dynamic response. With the magneto-electrically permeable assumption, both the magnetical and electrical impacts have no contribution to the crack tip field singularity. However, with the impermeable assumption, both the applied magnetical loads and electrical loads play a dominant role in the dynamic fracture behavior of crack tips. And for the two kinds of crack surface conditions, increasing the graded index can all retard the crack extension.     

Static response of functionally graded multilayered two-dimensional quasicrystal plates with mixed boundary conditions

Applied Mathematics and Mechanics - The unusual properties of quasicrystals (QCs) have attracted tremendous attention from researchers. In this paper, a semi-analytical solution is presented for...     

Three-dimensional exact analysis of a simply supported functionally gradient piezoelectric plate

An exact three-dimensional analysis is presented for a functionally gradient piezoelectric material rectangular plate that is simply supported and grounded along its four edges. The state equations of the functionally gradient piezoelectric material are developed based on the state space approach. Assuming that the mechanical and electric properties of the material have the same exponent-law dependence on the thickness-coordinate, we obtain an exact three-dimensional solution of the coupling electroelastic fields in the plate under mechanical, and electric loading on the upper and lower surfaces of the plate. The influences of the different functionally gradient material properties on the structural response of the plate to the mechanical and electric stimuli are then studied through examples.     

Direct approach-based analysis of plates composed of functionally graded materials

The classical plate theory can be applied to thin plates made of classical materials like steel. The first theory allowing the analysis of such plates was elaborated by Kirchhoff. But this approach was connected with various limitations (e.g., constant material properties in the thickness direction). In addition, some mathematical inconsistencies like the order of the governing equation and the number of boundary conditions exist. During the last century many suggestions for improvements of the classical plate theory were made. The engineering direction of improvements was ruled by applications (e.g., the use of laminates or sandwiches as the plate material), and so new hypotheses for the derivation of the governing equations were introduced. In addition, some mathematical approaches like power series expansions or asymptotic integration techniques were applied. A conceptional different direction is connected with the direct approach in the plate theory. This paper presents the extension of Zhilin’s direct approach to plates made of functionally graded materials. The second author was supported by DFG grant 436RUS17/21/07.     

Mode III crack problem in a functionally graded magneto-electro-elastic strip

Considering the material properties to be one-dimensionally dependent, this paper studied an anti-plane problem for an embedded crack and edge crack perpendicular to the boundary of a functionally graded magneto-electro-elastic strip. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the effects of the loading combination parameter, material gradient parameter and crack configuration on the field intensity factors and the energy release rates of the functionally graded magneto-electro-elastic strip.     

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.     

Three-dimensional analysis of transient thermal stresses in functionally graded plates

An analytical solution is presented for three-dimensional thermomechanical deformations of a simply supported functionally graded (FG) rectangular plate subjected to time-dependent thermal loads on its top and/or bottom surfaces. Material properties are taken to be analytical functions of the thickness coordinate. The uncoupled quasi-static linear thermoelasticity theory is adopted in which the change in temperature, if any, due to deformations is neglected. A temperature function that identically satisfies thermal boundary conditions at the edges and the Laplace transformation technique are used to reduce equations governing the transient heat conduction to an ordinary differential equation (ODE) in the thickness coordinate which is solved by the power series method. Next, the elasticity problem for the simply supported plate for each instantaneous temperature distribution is analyzed by using displacement functions that identically satisfy boundary conditions at the edges. The resulting coupled ODEs with variable coefficients are also solved by the power series method. The analytical solution is applicable to a plate of arbitrary thickness. Results are given for two-constituent metal-ceramic FG rectangular plates with a power-law through-the-thickness variation of the volume fraction of the constituents. The effective elastic moduli at a point are determined by either the Mori–Tanaka or the self-consistent scheme. The transient temperature, displacements, and thermal stresses at several critical locations are presented for plates subjected to either time-dependent temperature or heat flux prescribed on the top surface. Results are also given for various volume fractions of the two constituents, volume fraction profiles and the two homogenization schemes.     

Non-linear analysis of functionally graded plates under transverse and in-plane loads

Large deflection and postbuckling responses of functionally graded rectangular plates under transverse and in-plane loads are investigated by using a semi-analytical approach. Material properties are assumed to be temperature-dependent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The plate is assumed to be clamped on two opposite edges and the remaining two edges may be simply supported or clamped or may have elastic rotational edge constraints. The formulations are based on the classical plate theory, accounting for the plate-foundation interaction effects by a two-parameter model (Pasternak-type), from which Winkler elastic foundation can be treated as a limiting case. A perturbation technique in conjunction with one-dimensional differential quadrature approximation and Galerkin procedure are employed in the present analysis. The numerical illustrations concern the large deflection and postbuckling behavior of functional graded plates with two pairs of constituent materials. Effects played by volume fraction, the character of boundary conditions, plate aspect ratio, foundation stiffness, initial compressive stress as well as initial transverse pressure are studied.     

Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic plate containing an internal or an edge crack parallel to the graded direction is investigated. The crack is assumed to be magneto-electrically impermeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Field intensity factors and energy release rate are derived, analyzed and partially calculated numerically. The effects of material graded index, loading combination parameter (including size and direction) and geometry criterion of the plate on the dynamic energy release rate are shown graphically. Numerical results indicate that increasing the graded index can all retard the crack extension, and that both the applied magnetic field loadings and electric field loadings play a dominant role in the dynamic fracture behaviors of crack tips.     

Static analysis of functionally graded doubly-curved shells and panels of revolution

The Generalized Differential Quadrature (GDQ) Method is applied to study four parameter functionally graded and laminated composite shells and panels of revolution. The mechanical model is based on the so-called First-order Shear Deformation Theory (FSDT), in particular on the Toorani-Lakis Theory. The solution is given in terms of generalized displacement components of points lying on the middle surface of the shell. The generalized strains and stress resultants are evaluated by applying the Differential Quadrature rule to the generalized displacements. The transverse shear and normal stress profiles through the thickness are reconstructed a posteriori by using local three-dimensional elasticity equilibrium equations. In order to verify the accuracy of the present method, GDQ results are compared with the ones obtained with semi-analytical formulations and with 3D finite element method. A parametric study is performed to illustrate the influence of the parameters on the mechanical behavior of functionally graded shell structures made of a mixture of ceramics and metal.