Thermo-Electro-Mechanical Buckling of Shear Deformable Hybrid Circular Functionally Graded Material Plates

Buckling analysis of perfect circular functionally graded plates with surface-bounded piezoelectric layers based on the first-order shear deformation theory is presented in this article. The material properties of the functionally graded (FG) layer are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituents. The plate is assumed to be under constant electrical field and two types of thermal loadings, namely, the uniform temperature rise and nonlinear temperature gradient through the thickness. Also, the stability of a plate under radial mechanical compressive force is examined. The equilibrium and stability equations are derived based on the first-order shear deformation plate theory using a variational approach. The boundary condition of the plate as an immovable type of the clamped edge is considered. Resulting equations are employed to obtain the closed-form solution for the critical buckling temperature for each loading case. The effects of electric field, piezo-to-host thickness ratio, and power law index of functionally graded plates subjected to thermo-mechanical-electrical loads are investigated. The results are compared with the classical plate theory and verified with the available data in the open literature.     

Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations

This paper deals with free vibration analysis of radially functionally graded circular and annular sectorial thin plates of variable thickness, resting on the Pasternak elastic foundation. Differential quadrature method (DQM) is used to yield natural frequencies of the circular/annular sectorial plates under simply-supported and clamped boundary conditions on the basis of the classical plate theory (CPT). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is assumed to remain constant. The validity of the present solution is first examined by studying the convergence of the frequency parameters. Then, a comparison of results with those available in literature confirms the excellent accuracy of the present approach. Afterwards, the frequency parameters of the circular/annular sectorial thin plates with uniform, linear, and quadratic variations in thickness are computed for different boundary conditions and various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Elasticity solution of functionally graded circular and annular plates integrated with sensor and actuator layers using differential quadrature

Based on three-dimensional theory of elasticity axisymmetric static analysis of functionally graded circular and annular plates imbedded in piezoelectric layers is investigated using differential quadrature method (DQM). The plate has various edges boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. This method can give an analytical solution along the graded direction using the state space method (SSM) and an effective approximate solution along the radial direction using the one-dimensional DQM. The method is validated by comparing numerical results with the results obtained in the literature. Both the direct and the inverse piezoelectric effects are investigated and the influence of piezoelectric layers on the mechanical behavior of plate is studied. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the static behavior of FG circular and annular plates are investigated.     

Differential quadrature analysis of functionally graded circular and annular sector plates on elastic foundation

The main purpose of this study is to investigate buckling and free vibration behaviors of radially functionally graded circular and annular sector thin plates subjected to uniform in-plane compressive loads and resting on the Pasternak elastic foundation. In-plane compressive loads may be applied to either radial, circumferential, or all edges of circular/annular sector plates. Based on the classical plate theory (CPT), critical buckling loads and fundamental frequencies of the circular/annular sector plates under simply-supported and clamped boundary conditions are obtained by using differential quadrature method (DQM). The inhomogeneity of the plate is characterized by taking exponential variation of Young’s modulus and mass density of the material along the radial direction whereas Poisson’s ratio is considered to be constant. Convergence study is carried out to demonstrate the stability of the present method. To confirm the excellent accuracy of the present approach, a few comparisons are made for limited cases between the present results and those available in literature. Critical buckling load and fundamental frequency parameters of the circular/annular sector thin plates are computed for different boundary conditions, various values of the material inhomogeneity constants, sector angles, and inner to outer radius ratios.     

Nonlinear analysis of smart functionally graded annular sector plates using cylindrically orthotropic piezoelectric fiber reinforced composite

This work deals with the geometrically nonlinear thermo-electro-elastic analysis of functionally graded (FG) annular sector plates integrated with the annular patches of cylindrically orthotropic piezoelectric fiber reinforced composite (PFRC). The annular patches with an external voltage across their thickness act as the distributed actuators and their performance in controlling the nonlinear flexural deformations of the host FG plates is investigated. The temperature field is assumed to be spatially uniform over the plate surfaces and varied through the thickness of the substrate FG plates. The temperature-dependent material properties of the FG plates are assumed to be graded in the thickness direction of the plates according to a power-law distribution while the Poisson’s ratio is assumed to be a constant over the domain of the substrate plate. A finite element model of the overall smart FG annular sector plate is developed based on the first order shear deformation theory and the Von Karman nonlinear strain–displacement relations. The governing nonlinear finite element equations are derived employing the principle of minimum potential energy and solved using direct iteration method. The numerical results illustrate significant control authority of the cylindrically orthotropic PFRC annular patches for active control of nonlinear deformations of the substrate FG annular sector plates. The numerical results also reveal the best radial and circumferential locations of the annular PFRC patches for effective control. For a specified circumferential stretch of the annular PFRC patches, their minimum radial length is numerically estimated in such a way that the performance of the overall smart FG plate is not affected significantly. The effects of the material properties and the temperature of the host FG plate on the performance of the annular PFRC patches are also discussed.     

Hygrothermo-mechanical buckling of FGM plates resting on elastic foundations using a quasi-3D model

Hygrothermal and mechanical buckling responses of functionally graded (FG) plates resting on Winkler–Pasternak’s foundations are presented in this paper using a refined quasi-3D model. The effects due to transverse normal strain and shear deformation are both included. The present model exactly satisfies stress boundary conditions on the upper and lower surfaces of the FG plate without using shear correction factors. It is assumed that the material properties vary according to a power law of the thickness coordinate variable. The hygrothermal buckling equilibrium equations are derived from the principle of virtual work for FG plates resting on Winkler–Pasternak’s foundations with simply-supported boundary conditions. Two types of thermal and hygrothermal loading, uniform thermal and hygrothermal rise, linear thermal and hygrothermal distribution through the thickness are considered. Numerical results are presented to verify the accuracy of the present study. The effects played by Winkler–Pasternak’s parameters, plate aspect ratio, side-to-thickness ratio, gradient index, and loading type on the critical buckling of the FG plates are all investigated.     

Buckling analysis of a porous circular plate with piezoelectric sensor–actuator layers under uniform radial compression

This study presents the buckling analysis of a solid circular plate made of porous material bounded with piezoelectric sensor–actuator patches. The porous material properties vary through the thickness direction of the plate following a given function. The general mechanical nonlinear equilibrium and linear stability equations are derived using the variational formulations to obtain the governing equations of the piezoelectric porous plate. The buckling load is derived for solid circular plates under uniform radial compressive loading for the clamped edge condition. The effects of piezoelectric layers on the buckling load of the plate, piezoelectric layer-to-porous plate thickness ratio, feedback gain, and variation of porosity are investigated. The results are verified with the known results in the literature.     

Thermo elasticity solution of functionally graded,solid, circular,and annular plates integrated with piezoelectric layers using the differential quadrature method

Static analysis of functionally graded (FG) solid circular/annular plates imbedded in piezoelectric layers under thermo-electro mechanical load is investigated using the differential quadrature method. The plate has various edge boundary conditions and its material properties are assumed to vary in an exponential law with the Poisson ratio to be constant. The method is validated by comparing numerical results with the results obtained in the literature. The effects of the gradient index, thickness to radius ratio, and edges boundary conditions on the thermoelastic behavior of FG solid circular and annular plates are investigated.     

Free vibration analysis of functionally graded annular plates by state-space based differential quadrature method and comparative modeling by ANN

This paper deals with three-dimensional analysis of functionally graded annular plates through using state-space based differential quadrature method (SSDQM) and comparative behavior modeling by artificial neural network (ANN) for different boundary conditions. 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 used to obtain the vibration frequencies. The state variables include a combination of three displacement parameters and three stress parameters. Numerical results are given to demonstrate the convergency and accuracy of the present method. Once the semi-analytical method is validated, an optimal ANN is selected, trained and tested by the obtained numerical results. In addition to the quantitative input parameters, support type is also considered as a qualitative input in NN modeling. Eventually the results of SSDQM and ANN are compared and the influence of thickness of the annular plate, material property graded index and circumferential wave number on the non-dimensional natural frequency of annular functionally graded material (FGM) plates with different boundary conditions are investigated. The results show that ANN can acceptably model the behavior of FG annular plates with different boundary conditions.     

Experimental and numerical investigations of low velocity impact on functionally graded circular plates

This paper addresses low-velocity impact behaviour of functionally graded clamped circular plates. An experimental work was carried out to investigate the impact behaviour of FG circular plates which is composed of ceramic (SiC) and metal (Al) phases varying through the plate thickness by using a drop-weight impact test system. The influence of the compositional gradient exponent and impactor velocity on the contact forces and absorbed energies was concentrated on the tests. The explicit finite element method, in which a volume fraction based elastic–plastic model (the TTO model) was implemented for the functionally graded materials, was used to simulate their drop-weight impact tests. Effective material properties at any point inside FGM plates were determined using Mori–Tanaka scheme. The experimental and numerical results indicated that the compositional gradient exponent and impactor velocity more effective on the elasto-plastic response of the FG circular plates to a low-velocity impact loading. The comparison at the theoretical and experimental results showed that the use of the TTO model in modelling the elasto-plastic behaviour of FG circular plates results in increasing deviations between the numerical and experimental contact forces for ceramic-rich compositions whereas it becomes more successful for metal-rich compositions.     

A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations

The present paper investigates free vibration of variable thickness two-directional-functionally graded circular plates, resting on elastic foundations. The results are obtained for clamped, free, and simply supported edge conditions. Variations of the material and geometrical parameters are monitored by five distinct exponential functions. Therefore, the resulted non-dimensional solution may be used for a wide range of the practical problems. Mindlin’s plate theory and the differential transformation technique are used to obtain the governing equations of the natural frequencies of the circular plates. Effects of variations of the material properties in the radial and thickness directions, geometric parameters (e.g., the thickness-to-radius ratio in the center of the plate), stiffness parameters of the foundation, and various boundary conditions on the natural frequencies are investigated. Results reveal that by choosing a suitable combination of the material properties, the free vibration behavior of the thick plates may be enhanced without the need to change the geometric parameters.     

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.     

A 3-D Ritz solution for free vibration of circular/annular functionally graded plates integrated with piezoelectric layers

This paper addresses three-dimensional (3-D) free vibration characteristic of thick circular/annular functionally graded (FG) plates with surface-bonded piezoelectric layers on the basis of 3-D Ritz solution. Three displacement components along with electrical potential field of the plate are expressed by a set of Chebyshev polynomials multiplied by geometry boundary functions. Both open-circuit and closed-circuit surface conditions are taken into account. The mechanical properties of the FG plates are assumed to vary continuously through the thickness of the plate and obey either exponent or power law distribution of the volume fraction of the constituents. The effect of thickness-to-radius ratio, inner-to-outer radius ratio, piezo-to-host thickness ratio and gradient index on the natural frequencies of coupled piezoelectric FG circular/annular plates is investigated for different electrical and mechanical boundary conditions. It is observed that, unlike isotropic homogeneous circular/annular plates, frequency parameters of their piezoelectric coupled FG counterparts significantly increase with an enhancement in the host plate thickness to radius ratio. Results also show that the frequency parameters for open-circuit condition are higher than those for closed-circuit condition.     

Limit analysis of FGM circular plates subjected to arbitrary rotational symmetric loads using von-Mises yield criterion

In this paper, employing the limit analysis theorem, critical loading on functionally graded (FG) circular plate with simply supported boundary conditions and subjected to an arbitrary rotationally symmetric loading is determined. The material behavior follows a rigid-perfectly plastic model and yielding obeys the von-Mises criterion. In the homogeneous case, the highly nonlinear ordinary differential equation governing the problem is analytically solved using a variational iteration method. In other cases, numerical results are reported. Finally, the results are compared with those of the FG plate with Tresca yield criterion and also in the homogeneous case with those of employing the von-Mises yield criterion. A good correspondence is observed between the calculated results and those available in the literature.     

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.     

Thermal buckling analysis of functionally graded material beams

Buckling of beams made of functionally graded material under various types of thermal loading is considered. The derivation of equations is based on the Euler–Bernoulli beam theory. It is assumed that the mechanical and thermal nonhomogeneous properties of beam vary smoothly by distribution of power law across the thickness of beam. Using the nonlinear strain–displacement relations, equilibrium equations and stability equations of beam are derived. The beam is assumed under three types of thermal loading, namely; uniform temperature rise, nonlinear, and linear temperature distribution through the thickness. Various types of boundary conditions are assumed for the beam with combination of roller, clamped and simply-supported edges. In each case of boundary conditions and loading, a closed form solution for the critical buckling temperature for the beam is presented. The formulations are compared using reduction of results for the functionally graded beams to those of isotropic homogeneous beams given in the literature.     

Three-dimensional analytical solution for functionally graded magneto–electro-elastic circular plates subjected to uniform load

The problem of a functionally graded, transversely isotropic, magneto–electro-elastic circular plate acted on by a uniform load is considered. The displacements and electric potential are represented by appropriate polynomials in the radial coordinate, of which the coefficients depends on the thickness coordinate, and are called the generalized displacement functions. The governing equations as well as the boundary conditions for these generalized displacement functions are derived from the original equations of equilibrium for axisymmetric problems and the boundary conditions on the upper and lower surfaces of the plate. Explicit expressions are then obtained through a step-by-step integration scheme, with five integral constants determinable from the boundary conditions at the cylindrical surface in the Saint Venant’s sense. The analytical solution is suited to arbitrary variations of material properties along the thickness direction, and can be readily degenerated into those for homogeneous plates. A particular circular plate, with some material constants being the exponential functions of the thickness coordinate, is finally considered for illustration.     

Elastic response of functionally graded circular plates under a drop-weight

The three-dimensional impact behaviours of functionally graded (FG) circular plates were studied under a drop-weight. The functionally graded circular plate was composed of ceramic (SiC) and metal (Al) phases and the through-thickness mechanical properties through the region between the metal and ceramic layers vary continuously according to a power-law distribution of the volume fraction of the ceramic. The through-thickness material properties of the FG circular plate were determined using the Mori–Tanaka scheme. The effects of layer number and compositional gradient exponent as well as impactor velocity and plate radius on the elastic impact response of the FG circular plates were investigated. The compositional gradient exponent, impactor velocity and plate radius played an important role on the impact response of the FG circular plates, whereas the layer number through the plate thickness had a minor effect. In addition, the failure strains in all layers were determined using Tamura–Tomota–Ozowa (TTO) model in order to predict the damage regions in each layer through the thickness of FG circular plates.     

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.     

Thermal buckling and postbuckling analysis of functionally graded annular plates with temperature-dependent material properties

As a first endeavor, the thermal buckling and postbuckling analysis of functionally graded (FG) annular plates with material properties graded in the radial direction is presented. The formulation is derived based on the first-order shear deformation theory (FSDT) and the geometrical nonlinearity is modeled using Green’s strain in conjunction with von Karman’s assumptions. The material properties are temperature-dependent and graded according to the power law distribution. It is assumed that the temperature varies along the radial direction. Using the virtual work principle, the pre-buckling and postbuckling equilibrium equations and the related boundary conditions are derived. Differential quadrature method (DQM) as an efficient numerical technique is adopted to solve the governing equations. The presented formulation and the method of solution are validated by performing convergence and comparison studies with available results in the literature. Finally, the effects of volume fraction index, geometrical parameters, mechanical/thermal properties of the constituent materials and boundary conditions on the thermal buckling and postbuckling behavior of the radially graded annular plate are evaluated and discussed.