Thermodynamic effect on the bending response of elastic foundation FG plate by using a novel four variable refined plate theory

This article deals with the study of the thermodynamic behavior of functionally graded material plates resting on two-parameter elastic foundation. An analytical solution based on a new shear refined deformation theory is presented. The displacement field used in the present refined theory contains undetermined integral forms and involves only four unknowns to derive. The plate is assumed simply supported and subjected to two different temperatures fields across its thickness. The mechanical characteristics of the plate are assumed to be varied across the thickness according to a simple exponential law distribution. The governing equations and boundary conditions are derived using the principle of virtual displacements and Navier solution technique is adopted to derive analytical solutions. A detailed numerical study of the present new refined theory is carried out to examine the influence of the time’s parameter, foundation’s parameters and deflection on the bending response of the FG plate.     

Two-Variable Refined Plate Theory for Thermoelastic Bending Analysis of Functionally Graded Sandwich Plates

The thermoelastic bending analysis of functionally graded sandwich plates using the two-variable refined plate theory is presented in this paper. Unlike any other theory, the number of unknown functions involved is only four, as against five in case of other shear deformation theories. The theory presented is variationally consistent, has strong similarity with classical plate theory in many aspects, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity, Poisson's ratio of the faces, and thermal expansion coefficients are assumed to vary according to a power law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. The influences played by the transverse shear deformation, thermal load, plate aspect ratio, and volume fraction distribution are studied. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated. It can be concluded that the proposed theory is accurate and simple in solving the thermoelastic bending behavior of functionally graded plates.     

Thermal Buckling of Functionally Graded Plates Resting On Elastic Foundations Using the Trigonometric Theory

In this article, thermal buckling analysis of functionally graded material (FGM) plates resting on two-parameter Pasternak's foundations is investigated. Equilibrium and stability equations of FGM plates are derived based on the trigonometric shear deformation plate theory and includes the plate foundation interaction and thermal effects. The material properties vary according to a power law form through the thickness coordinate. The governing equations are solved analytically for a plate with simply supported boundary conditions and subjected to uniform temperature rise and gradient through the thickness. Resulting equations are employed to obtain the closed-form solution for the critical buckling load for each loading case. The influences of the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the buckling temperature difference are discussed.     

Elasticity Solution for Bending Response of Functionally Graded Sandwich Plates Under Thermomechanical Loading

The thermomechanical bending response of functionally graded sandwich plates has been investigated by the use of the new four variable refined plate theories. The plate properties are assumed to be varied through the thickness following a simple power law distribution in terms of volume fraction of material constituents. The theory presented is variationally consistent, does not require shear correction factor, and gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free surface conditions. The no symmetric sandwich plate faces are made of isotropic, two-constituent (ceramic–metal) material distribution through the thickness. The core layer is still homogeneous and made of an isotropic metal material. Several kinds of no symmetric sandwich plates are presented. The validity of the present theory is investigated by comparing some of the present results with those of the classical, the first-order, and the other higher-order theories. Field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory are derived. Displacement and stress functions of the plate for different values of the power-law exponent and thickness to-side ratios are presented. Numerical results for deflections and stresses of functionally graded metal–ceramic plates are investigated.     

Discrete singular convolution method for the analysis of Mindlin plates on elastic foundations

The method of discrete singular convolution (DSC) is used for the bending analysis of Mindlin plates on two-parameter elastic foundations for the first time. Two different realizations of singular kernels, such as the regularized Shannon's delta (RSD) kernel and Lagrange delta sequence (LDS) kernel, are selected as singular convolution to illustrate the present algorithm. The methodology and procedures are presented and bending problems of thick plates on elastic foundations are studied for different boundary conditions. The influence of foundation parameters and shear deformation on the stress resultants and deflections of the plate have been investigated. Numerical studies are performed and the DSC results are compared well with other analytical solutions and some numerical results.     

Thermal buckling analysis of functionally graded sandwich plates

Thermal buckling of functionally graded sandwich plates are presented in this article. Two common types of FGM sandwich plates, namely, homogeneous face layers with FGM core and FGM face layers with homogeneous core are considered. Material properties and thermal expansion coe?cient of FGM layers are assumed to vary continuously through-the-thickness according to a simple power-law distribution in terms of the volume fractions of the constituents. Equilibrium and stability equations of FGM sandwich plate with simply supported boundary conditions are derived using the higher-order shear deformation plate theory. The influence of the plate aspect ratio, the relative thickness, the gradient index, and the thermal loading conditions on the critical buckling temperature of FGM sandwich plates are investigated. The thermal loads are assumed to be uniform, linear, and nonlinear distribution through-the-thickness. A new simple solution for thermal buckling of FGM sandwich plates under nonlinear temperature rise is presented.     

Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution

This paper presents nonlinear static analysis of a rectangular laminated composite thick plate resting on nonlinear two-parameter elastic foundation with cubic nonlinearity. The plate formulation is based on first-order shear deformation theory (FSDT). The governing equation of motion for a rectangular laminated composite thick plate is derived by using the von Karman equation. The nonlinear static deflections of laminated plates on elastic foundation are investigated using the discrete singular convolution method. The effects of foundation and geometric parameters of plates on nonlinear deflections are investigated. The validity of the present method is demonstrated by comparing the present results with those available in the literature.     

Thermal stability of eccentrically stiffened FGM plate on elastic foundation based on Reddy's third-order shear deformation plate theory

This article studies the nonlinear thermal buckling and postbuckling of eccentrically stiffened functionally graded plates on elastic foundation subjected to mechanical, thermal, and thermomechanical loads. The noticeable point of this study is using the Reddy's higher order shear deformation plate theory and a general formula for the forces and moments of eccentrically stiffened functionally graded material (FGM) plate, which takes into account the influence of temperature on both the FGM plate and stiffeners. The article used the Galerkin method, stress function, and iterative method to determine the thermal buckling loads and postbuckling response of the eccentrically stiffened FGM plates in three different cases of boundary conditions. The effects of material, temperature-dependent material properties, elastic foundations, boundary conditions, outside stiffeners, and temperature on the buckling and postbuckling loading capacity of the FGM plates in thermal environments are analyzed and discussed. A good agreement is obtained by comparing the present analysis with other available literature.     

Mechanical Behavior of Rectangular Plates with Functionally Graded Coefficient of Thermal Expansion Subjected to Thermal Loading

This study analyzed an elastic, rectangular, and simply supported functionally graded material (FGM) plate with medium thickness subjected to linear temperature change in the z direction. Young's modulus and Poisson ratio of the FGM plates are assumed to remain constant throughout the entire plate. However, the coefficient of thermal expansion of the FGM plate varies continuously throughout the thickness direction in relation to the volume fraction of constituents defined by power-law, sigmoid, or exponential functions. The series solutions for the power-law FGM (P-FGM), sigmoid FGM (S-FGM), or exponential FGM (E-FGM) plates subjected to thermal loading are obtained based on the classical plate theory and Fourier series expansion. The analytical solutions for P-, S-, and E-FGM plates are verified by numerical results obtained with the finite element technique.     

Thermal Buckling of Functionally Graded Plates According to a Four-Variable Refined Plate Theory

In this article, a four-variable refined plate theory is presented for buckling analysis of functionally graded plates. The theory, which has strong similarity with classical plate theory in many aspects, accounts for a quadratic variation of the transverse shear strains across the thickness and satisfies the zero traction boundary conditions on the top and bottom surfaces of the plate without using shear correction factors. A power law distribution is used to describe the variation of volume fraction of material compositions. Equilibrium and stability equations are derived based on the present theory. The non-linear governing equations are solved for plates subjected to simply supported boundary conditions. The thermal loads are assumed to be uniform, linear and non-linear distribution through-the-thickness. The influences of many plate parameters on buckling temperature difference will be investigated. It is noticed that the present refined plate theory can predict accurately the critical temperatures of simply supported functionally graded plates.     

Thermal Response of Symmetric Cross-Ply Laminated Plates Subjected to Linear and Non-Linear Thermo-Mechanical Loads

The flexural response of symmetric cross-ply laminated plates subjected to uniformly distributed linear and non-linear thermo-mechanical loads is presented using trigonometric shear deformation theory. The in-plane displacement field uses sinusoidal function in terms of thickness coordinate to include the shear deformation effect. The theory satisfies the shear stress-free boundary conditions on the top and bottom surfaces of the plate. The present theory obviates the need of shear correction factor. Governing equations and boundary conditions of the theory are obtained using the principle of virtual work. Thermal stresses and displacements for three-layer symmetric square cross-ply laminated plates subjected to uniform linear and nonlinear and thermo-mechanical loads are obtained. The results of present theory are compared with those of classical plate theory, first-order shear deformation theory and higher-order shear deformation theory.     

Exact Solution for Thermal Buckling of Functionally Graded Plates Resting on Elastic Foundations with Various Boundary Conditions

In this article, we investigate the buckling analysis of plates that are made of functionally graded materials (FGMs) resting on two-parameter Pasternak's foundations under thermal loads. Three different thermal loads were considered, i.e., uniform temperature rise (UTR), linear and non-linear temperature distributions (LTD and NTD) through the thickness. The mechanical and thermal properties of functionally graded material (FGM) vary continuously along the plate thickness according to a simple power law distribution. Employing an analytical approach, the five coupled governing stability equations, which are derived based on first-order shear deformation plate theory, are converted into two uncoupled partial differential equations (PDEs). Considering the Levy-type solution, these two PDEs are reduced to two ordinary differential equations (ODEs) with variable coefficients. Then, the ODEs are solved using an exact analytical solution, which is called the power series Frobenius method. The appropriate convergence study and comparison with previously published related articles was employed to verify the accuracy of the proposed method. After such verifications, the effects of parameters such as the plate aspect ratio, side-to-thickness ratio, gradient index, and elastic foundation stiffnesses on the critical buckling temperature difference are illustrated and explained. The critical buckling temperatures of functionally graded rectangular plates with six various boundary conditions are reported for the first time and can serve as benchmark results for researchers to validate their numerical and analytical methods in the future.     

Nonlinear thermal dynamic response of shear deformable FGM plates on elastic foundations

This paper investigates the nonlinear dynamic response of thick functionally graded materials (FGM) plates using the third-order shear deformation plate theory and stress function. The FGM plate is assumed to rest on elastic foundations and subjected to thermal and damping loads. Numerical results for dynamic response of the FGM plate are obtained by Runge–Kutta method. The results show the influences of geometrical parameters, the material properties, the elastic foundations, and thermal loads on the nonlinear dynamic response of FGM plates.     

Nonlinear dynamic response and vibration of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) shear deformable plates with temperature-dependent material properties and surrounded on elastic foundations

Based on Reddy’s third-order shear deformation plate theory, the nonlinear dynamic response and vibration of imperfect functionally graded carbon nanotube-reinforced composite (FG-CNTRC) plates on elastic foundations subjected to dynamic loads and temperature are presented. The plates are reinforced by single-walled carbon nanotubes which vary according to the linear functions of the plate thickness. The plate’s effective material properties are assumed to depend on temperature and estimated through the rule of mixture. By applying the Airy stress function, Galerkin method and fourth-order Runge–Kutta method, nonlinear dynamic response and natural frequency for imperfect FG-CNTRC plates are determined. In numerical results, the influences of geometrical parameters, elastic foundations, initial imperfection, dynamic loads, temperature increment, and nanotube volume fraction on the nonlinear vibration of FG-CNTRC plates are investigated. The obtained results are validated by comparing with those of other authors.     

Nonlinear dynamic response of eccentrically stiffened FGM plate using Reddy’s TSDT in thermal environment

The nonlinear dynamics of an eccentrically stiffened functionally graded material (ES-FGM) plates resting on the elastic Pasternak foundations subjected to mechanical and thermal loads is considered in this article. The plates are reinforced by outside stiffeners with temperature-dependent material properties in two cases: uniform temperature rise and through the thickness temperature gradient. Both stiffeners and plate are deformed under temperature. Using Reddy’s third-order shear deformation plate theory, stress function, Galerkin and fourth-order Runge–Kutta methods, the effects of material and geometrical properties, temperature-dependent material properties, elastic foundations, and stiffeners on the nonlinear dynamic response of the ES-FGM plate in thermal environments are studied and discussed. Some obtained results are validated by comparing with those in the literature.     

A Four-Variable Plate Theory for Thermoelastic Bending Analysis of Laminated Composite Plates

In this article, the thermoelastic bending analysis of laminated composite plates subjected to thermal load linear across the thickness using the four variable refined plate theory is presented. The theory involves four unknown variables, as against five in case of other higher-order theories and first-order shear deformation theory. The theory gives rise to transverse shear stress variation such that the transverse shear stresses vary parabolically across the thickness satisfying shear stress free conditions at top and bottom surfaces of the plate. The theory does not require problem-dependent shear correction factors that are associated with the first-order shear deformation theory. The principle of virtual work is used to obtain variationally consistent governing equations and boundary conditions. The simply supported laminated composite plates are considered for the detail numerical study. A closed-form solution is obtained using the double trigonometric series technique suggested by Navier. The numerical results for thermal displacements and stresses of laminated composite plates are obtained and compared with those of other refined theories and exact solutions wherever applicable to assess the efficiency of the present theory. From the numerical results it is observed that since plate is subjected to pure thermal load and not subjected to transverse mechanical load, the present theory has strong similarity with classical plate theory in many aspects.     

Research on shakedown analysis for functionally graded material plate under thermomechanical loading

Shakedown is an important problem in the design and analysis of functionally graded structures subjected to cyclic, thermal, and mechanical loadings. Subjected to constant mechanical load and cyclic temperature change, static shakedown of a functionally graded material plate and its homogenous counterpart were analyzed in this article with the approach proposed by the authors previously. The functionally graded material plate is composed of an elastoplastic matrix Al and elastic particles SiC, and the particle volume fraction varies through the thickness. The distributions of the effective mechanical and thermal properties of the composites through the thickness are described graded continuously with an exponential law. The results show that a proper and continuously graded distribution of material properties can efficiently improve the shakedown capability of the functionally graded material plate and also show the significance of shakedown analysis and application for functionally graded material plates.     

Hygrothermomechanical Effects on Laminated Composite Plates in Terms of a Higher-Order Global-Local Model

Results analytical or numerical on transverse shear stresses of laminated composite plates subjected to hygrothermomechanical effects are scarce in literature. To fill this gap, a higher-order global-local model (HGLM) satisfying the continuity conditions of transverse shear stresses at interfaces is proposed. Based on stress continuity condition between layers, the number of variables in the proposed model is independent of the number of layers of the laminate. Applying Navier's technique to equilibrium equations obtained using the principle of minimum potential energy, analytical solution of the model HGLM is derived for simply supported composite plates. Comparing the results from available three-dimensional elasticity theory and those computed from the first-order and the higher-order models, it is found that the proposed model can produce promising transverse shear stresses directly from constitutive equations without any smoothing technique. The effects of material properties, aspect ratio, side-to-thickness ratio, stacking sequence and thermal expansion coefficients on the hygrothermomechanical response have also been studied.     

REFINED THEORY FOR THERMOELASTIC STABILITY OF FUNCTIONALLY GRADED CIRCULAR PLATES

Axisymmetric thermal and mechanical buckling of functionally graded circular plates is considered. Equilibrium and stability equations under thermal and mechanical loads are derived based on first-order shear deformation 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 ordinary differential equations is established. Buckling analysis of a functionally graded plate under uniform temperature rise, linear and nonlinear gradient through the thickness, and uniform radial compression are considered, and the critical buckling loads are derived for clamped edge plates. The results are compared with the buckling loads obtained for a functionally graded plate based on the classical plate theory given in the literature.     

Thermal Effects on Stress Analysis in Large Amplitude Vibration of a Thick Annular Functionally Graded Plate

This paper deals with the nonlinear free and forced vibration of thick annular functionally graded material plates. The temperature field considered is assumed to be a uniform distribution over the plate surface and varied in the thickness direction only. Material properties are assumed to be temperature-dependent, and graded in the thickness direction. The formulations are based on the first-order shear deformation plate theory and von Kármán-type equation. The numerical illustrations concern with nonlinear vibration characteristics of functional graded plates with two constituent materials in thermal environments. Effects of material compositions and thermal loads on the vibration characteristics and stresses are examined.