Functionally Graded Material Circular Plate Research Articles

Active control of thermally induced vibrations of temperature-dependent FGM circular plate with piezoelectric sensor/actuator layers

This study investigates the novel application of active control methods in mitigating thermally induced vibrations of functionally graded material (FGM) circular plates subjected to thermal shock. The top surface of the FGM circular plate is exposed to rapid surface heating, resulting in thermally induced vibrations. To counteract these deflections, two piezoelectric layers, one functioning as a sensor and the other as an actuator, are strategically integrated into the plate opposite sides. This layered configuration effectively distances the direct rapid surface heating from piezoelectric layers. All thermos-electro-mechanical properties of the used materials are considered temperature-dependent. The research employs the Generalized differential quadrature element (GDQE) method and the Crank-Nicolson time marching procedure to solve the nonlinear Fourier-type heat conduction equations for the layered structure. After obtaining the transient temperature profile, the study calculates thermal moments, forces, and pyroelectric terms. These thermally induced effects are then integrated into the motion and electrostatic equations, which inherently exhibit nonlinearity due to the von Kármán large deformations theory. The plate displacements are approximated using the first-order shear deformation theory (FSDT) along the plate thickness, while a quadrature distribution is considered for approximating the electric potential within the piezoelectric layers. The highly coupled electro-mechanical equations are solved using the GDQ method and the Newmark approach, with the Newton-Raphson iterative scheme utilized to address nonlinearity. A significant contribution of this research lies in the exploration of three distinct active control strategies: proportional control, derivative control, and proportional-derivative control. The study thoroughly analyzes the impact of each control type on the plate response, considering various boundary conditions.

Postbuckling responses of porous FGM spherical caps and circular plates including edge constraints and nonlinear three-parameter elastic foundations

An analytical investigation on nonlinear stability of porous functionally graded spherical caps and circular plates under uniform external pressure and elevated temperature, respectively, is presented in this article. Porosities are evenly and unevenly distributed within the structures and the effective properties of functionally graded material (FGM) are determined according to a modified rule of mixture. Governing equations are established within the framework of classical theory taking into account geometrically nonlinear terms, initial geometric imperfection, and interactive pressure from nonlinear three-parameter elastic foundation. Both temperature dependence of material properties and tangential elasticity of edge constraint are included. Approximate analytical solutions are assumed to satisfy clamped condition of boundary edge and Galerkin method is applied to derive load-deflection relations from which buckling loads and postbuckling paths are determined. Numerical results indicate that porosities have deteriorative and beneficial influences on load-carrying capacity of structures under external pressure and elevated temperature, respectively. Furthermore, the snap-through buckling response of pressurized FGM spherical cap can be alleviated and even eliminated by virtue of elastic foundations with suitable stiffness parameters.

Nonlinear axisymmetric thermomechanical response of FGM circular plates

Based on von Karman plate theory, axisymmetric geometrically nonlinear governing equations of simple power-law type functionally graded material (FGM) circular plates are derived. Material properties are assumed to be temperature-independent, and thermal or/and mechanical loads are considered in the derivation. Considering clamped boundary conditions, the dimensionless critical (lowest) buckling temperature (temperature difference) 14.684 of the systems is obtained by analyzing linear-eigenvalue problem. With this constant and an analytical formula proposed in this paper, accurate dimensional critical (lowest) buckling temperature (temperature difference) of any specified clamped FGM circular plate can be easily calculated. Moreover, two-point boundary value problem posed by the governing equations and the clamped boundary conditions is solved using the shooting method. Thermal buckling response under thermal load and geometrically nonlinear mechanical behaviors under thermomechanical load of the system are discussed. When the temperature difference of the upper and lower surfaces of FGM plates exists, the solution of one-dimension Fourier heat conduction equation is an infinite series, which depicts a nonlinear temperature field (NLTF) along the thickness direction of plates. The effects of the initial terms of the series on the accuracy of solutions are examined. The results reveal it is necessary to take enough the initial terms of the series even for thinner FGM plates. Typical load–deflection curves of the clamped FGM circular plates with different temperature fields (along the thickness) are presented. For a given temperature field, the clamped FGM circular plates under transverse uniform load are a hard stiffness nonlinear system. When the values of thermal and mechanical load and their loading sequence meet certain combinations, the clamped circular plate will inevitably undergo secondary buckling (snap-through buckling).

Thermoelastic damping in functionally graded material circular micro plates

In this article, thermoelastic damping (TED) in functionally graded material (FGM) circular micro plates is analyzed based on classical plate theory and one-way coupled heat conduction equation. The material properties are assumed to be varied continuously in the thickness direction of the plate. A one-way coupled heat conduction equation with variable coefficients is solved by using a layer-wise homogenization approach. The complex frequency, including TED, of the FGM micro plate is obtained in terms of the natural frequency of the corresponding isothermal homogenous plate without TED from the mathematical similarity between the eigenvalue problems of governing differential equations of the two kind structures. Numerical results of TED for a ceramic-metal composite FGM circular micro plate are presented quantitatively to show the effects of the material gradient index, the geometry, the vibration mode shapes, and the environmental temperature on the TED in detail. The present mathematical model and solution method can be also used to evaluate TED in the FGM circular and annular plates with other through-thickness material gradient forms.

Thermal buckling and postbuckling of FGM circular plates with in-plane elastic restraints

Based on von Karman’s plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material (FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.

Statics of FGM circular plate with magneto-electro-elastic coupling: axisymmetric solutions and their relations with those for corresponding rectangular beam

This paper investigates the static behavior of a functionally graded circular plate made of magneto-electro-elastic (MEE) materials under tension and bending. The analysis is directly based on the three-dimensional governing equations for magneto-electro-elasticity, with the boundary conditions on the upper and lower surfaces satisfied exactly and those on the cylindrical surface satisfied approximately (in the Saint Venant sense). The analytical solutions, derived with a direct displacement method, are valid for any functionally graded material (FGM) with its properties varying independently in a continuous manner along the thickness direction. For homogeneous materials, these solutions are degenerated to the ones available in the literature. Interesting relations are also found between the solutions for a functionally graded magneto-electro-elastic (FGMEE) circular plate and those for an FGMEE rectangular beam, and even those for a functionally graded elastic beam when only the elastic displacements are considered. The beam solutions are also derived using a direct displacement method. Numerical examples are presented to verify the present analytical solutions, show the effects of material heterogeneity and multi-field coupling, and indicate the correspondence between the plate solutions and beam solutions.

Thermal Postbuckling of Imperfect Circular Functionally Graded Material Plates: Examination of Voigt, Mori–Tanaka, and Self-Consistent Schemes

Thermal postbuckling of solid circular plates made of a through-the-thickness functionally graded material (FGM) is analyzed in this paper. Initial imperfection of the plate is also taken into account. Each thermomechanical property of the plate is assumed to be a function of the temperature and thickness coordinate. Equivalent properties of the FGM media are obtained based on three different homogenization schemes, namely, Voigt rule, Mori–Tanaka scheme, and self-consistent estimate. Temperature profile is assumed to be through-the-thickness direction only. The solution of the heat conduction equation is obtained using an iterative central finite difference scheme. Various types of thermal loadings, such as uniform temperature rise, temperature specified at surfaces, and heat flux, are considered. Nonlinear equilibrium equations of the plate are obtained by means of the conventional Ritz method. Solution of the resulting nonlinear equilibrium equations and temperature distribution are obtained simultaneously at each step of heating. It is shown that response of a perfect clamped FGM plate is of the bifurcation type of buckling with stable postbuckling equilibrium branch, whereas imperfect clamped and perfect/imperfect simply supported FGM plates do not reveal the bifurcation type of instability through the nonuniform heating process. Furthermore, amplitude of initial imperfection is an important factor on the equilibrium path of FGM circular plates, especially for simply supported ones.

Thermal Post-Buckling of Functionally Graded Material Circular Plates Subjected to Transverse Point-Space Constraints

Axisymmetrical thermal post-buckling of functionally graded material (FGM) circular plates with immovably clamped boundary and a transversely central point-space constraint was studied. The material properties of the plate were assumed to vary as power law functions in the thickness direction and the temperature rise field to change only in the thickness direction. Based on von Karman's non-linear plate theory, governing equations in terms of the displacements of the middle plane were established. Temperature rise field was obtained by solving the one-dimensional heat conduction equation associated with specified boundary conditions at the top and bottom surface of the plate. By using the shooting method, thermal post-buckling deformation of the FGM circular plate was obtained before and after the plate contacting the point-space constraint. The changes in the characteristics of the deformation and the internal forces of FGM plates were discussed. The effects of gradients of material properties and non-uniform temperature rise parameters on the thermal post-buckling behaviors of FGM circular plates were also examined.

Nonlinear thermomechanical stability of shear deformable FGM shallow spherical shells resting on elastic foundations with temperature dependent properties

This paper presents an analytical approach to investigate the nonlinear stability of clamped functionally graded material (FGM) shallow spherical (SS) shells and circular plates resting on elastic foundations, subjected to uniform external pressure and exposed to thermal environments. 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 constituents. Formulations for axisymmetrically deformed SS shells are based on the first order shear deformation theory taking geometrical nonlinearity, initial geometrical imperfection and interaction of Pasternak type elastic foundations into consideration. Approximate solutions are assumed to satisfy clamped immovable boundary conditions and Galerkin method is applied to derive expressions of buckling loads and load–deflection curves for FGM SS shells. Specialization of these expressions gives corresponding relations of FGM circular plates, and an iterative algorithm is adopted to obtain buckling temperatures and postbuckling temperature–deflection curves for thermally loaded FGM circular plates. The effects of material, geometry and foundation parameters, imperfection and temperature dependence of material properties on the nonlinear response of FGM SS shells and circular plates are analyzed and discussed in detail.

Thermoelastic Transient Behavior for a Clamped FGM Circular Plate

In this paper, the thermoelastic transient behavior of a clamped circular plate composed of functionally graded material (FGM) is investigated. The material properties of the FGM circular plate are assumed to vary through the plate thickness according to a power law distribution of the volume fraction of constituent materials, except Poisson's ratio, which is assumed as constant. Based on the von Karman equation and classical theory of thin plates, the equation of motion for the FGM circular plate is derived by the Hamilton principle. The nonlinear governing equation is solved by the Galerkin method, along with Newmark's integration method, in an iterative manner. Numerical results reveal that the functional gradient index, ratio of thickness to radius, thermal and mechanical loads have significant effect on the thermoelastic transient behavior of the clamped FGM circular plate. The result presented herein may be used as a reference for solving other transient coupled problems of thermoelasticity.

Nonlinear Dynamic Analysis for FGM Circular Plates

AbstractIn the paper, nonlinear dynamic analysis of a circular plate composed of functionally graded material (FGM) is presented. Considering a transverse shear deformation and geometric nonlinearity, the von Karman geometric relation of the FGM circular plate is established. Based on the theory of the first-order shear deformation, a new set of equilibrium equations is developed by the principle of minimum total energy. Applying the finite difference method and Newmark scheme, the nonlinear transient problem is solved by the iterative method. To validate the presented method, the transient problem of the FGM circular plate is compared with those of the existed literature, and good agreement is observed. The effects of the volume fraction index, boundary conditions, mechanical load and the ratio of thickness to radius on the nonlinear transient problem of the FGM circular plate are investigated.

A power series solution for vibration and complex modal stress analyses of variable thickness viscoelastic two-directional FGM circular plates on elastic foundations

In the present paper, a power series solution is developed for free vibration and damping analyses of viscoelastic functionally graded plates with variable thickness on elastic foundations. It is assumed that the material properties of the functionally graded material (FGM) vary in the transverse and radial directions, simultaneously. Therefore, the presented solution can be employed for the transversely-graded and radially-graded viscoelastic circular plates, as special cases. In addition to the edge conditions, the plate may be resting on a two-parameter elastic foundation. The complex modulus approach in combination with the elastic–viscoelastic correspondence principle is employed to obtain the solution for various edge conditions. A sensitivity analysis including effects of various edge conditions, geometric parameters, coefficients of the elastic foundation, parameters of the functionally graded material, and material loss factor is carried out. In the present paper, concept of the complex modal stresses of the viscoelastic plates is discussed in detail.

Exact Relationships between Eigenvalues of FGM Circular Plates Based on RPT and those of Isotropic Homogeneous Circular Plates Based on CPT

Based on the mathematical similarity of the eigenvalue problem of the Reddy’s third-order plate theory (RPT) and the classical plate theory (CPT), relationships between the solutions of axisymmetric vibration or buckling of functionally graded material (FGM) circular plates based on RPT and those of isotropic homogeneous circular plates based on CPT are presented, from which one can easily obtain the RPT solutions of axisymmetric vibration or buckling of FGM circular plates expressed in terms of the well-known CPT solutions of isotropic circular plates without much tedious mathematics. Effects of rotary inertia are not considered in the present analysis. The relationships obtained from the present analysis may be used to check the validity, convergence and accuracy of numerical results of FGM plates based on RPT, and also show clearly the intrinsic features of the effect of transverse shear deformation on the classical solutions.

Nonlinear thermomechanical post-buckling of circular FGM plate with geometric imperfection

Nonlinear thermomechanical post-buckling of an imperfect functionally graded material (FGM) circular plate, subjected to both mechanical load and transversely non-uniform temperature rise, is presented. The material properties of FGM plates are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on von Kármán's plate theory, equilibrium equations governing a large axi-symmetric deformation of the FGM circular plate under thermomechanical loads are derived. In the analysis, the geometric imperfections of the plate are taken into account. By using a shooting method the nonlinear ordinary differential equations with immovably clamped boundary conditions are solved numerically. Responses for the nonlinear thermomechanical post-buckling responses of the FGM plate are obtained. Numerical examples are presented that relate to the performances of perfect and imperfect, homogenous and graded plates. Characteristic curves of the post-buckling deformation of the imperfect FGM circular plate varying with thermal loads, imperfection parameters and volume fraction index are plotted. And then effects of the load parameters, materials constitution, and the geometric imperfection of the plate on the deformation are discussed in detail.

Thermal buckling of functionally graded circular plates based on higher order shear deformation plate theory

In this research, thermal buckling of circular plates compose of functionally graded material (FGM) is considered. Equilibrium and stability equations of a FGM circular plate under thermal loads are derived, based on the higher order shear deformation plate theory (3rd order plate theory). Assuming that the material properties vary as a power form of the thickness coordinate variable z and using the variational method, the system of fundamental partial differential equations is established. A buckling analysis of a functionally graded circular plate (FGCP) under various types of thermal loads is carried out and the result are given in closed-form solutions. The results are compared with the critical buckling temperature obtained for FGCP based on first order (1st order plate theory) and classical plate theory (0 order plate theory) given in the literature. The study concludes that higher order shear deformation theory accurately predicts the behavior of FGCP, whereas the first order and classical plate theory overestimates buckling temperature.