Nonlinear analysis of sandwich plates with FGM face sheets resting on elastic foundations

Nonlinear vibration, nonlinear bending and postbuckling analyses are presented for a sandwich plate with FGM face sheets resting on an elastic foundation in thermal environments. The material properties of FGM face sheets 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. The governing equation of the plate that includes plate-foundation interaction is solved by a two-step perturbation technique. The thermal effects are also included and the material properties of both FGM face sheets and homogeneous core layer are assumed to be temperature-dependent. The numerical results reveal that the foundation stiffness and temperature rise have a significant effect on the natural frequency, buckling load, postbuckling and nonlinear bending behaviors of sandwich plates. The results also reveal that the core-to-face sheet thickness ratio and the volume fraction distribution of FGM face sheets have a significant effect on the natural frequency, buckling load and postbuckling behavior of the sandwich plate, whereas this effect is less pronounced for the nonlinear bending, and is marginal for the nonlinear to linear frequency ratios of the same sandwich plate.     

Nonlinear bending and postbuckling of FGM cylindrical panels subjected to combined loadings and resting on elastic foundations in thermal environments

A nonlinear analysis is presented for FGM cylindrical panels resting on elastic foundations subjected to the combined actions of uniform lateral pressure and compressive edge loads in thermal environments. The two cases of postbuckling of initially pressurized FGM cylindrical panels and of nonlinear bending of initially compressed cylindrical panels are considered. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori-Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel-foundation interaction and thermal effects are also included. The governing equations are solved by a singular perturbation technique along with a two-step perturbation approach. The numerical illustrations concern the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels with two constituent materials resting on Pasternak elastic foundations. The effects of volume fraction index, temperature variation, foundation stiffness as well as initial stress on the postbuckling behavior and the nonlinear bending response of FGM cylindrical panels are discussed in detail.     

Nonlinear vibration of shear deformable FGM cylindrical shells surrounded by an elastic medium

This paper investigates the large amplitude vibration behavior of a shear deformable FGM cylindrical shell of finite length embedded in a large outer elastic medium and in thermal environments. The surrounding elastic medium is modeled as a Pasternak foundation. Two kinds of micromechanics models, namely, Voigt model and Mori-Tanaka model, are considered. The motion equations are based on a higher order shear deformation shell theory that includes shell-foundation interaction. The thermal effects are also included and the material properties of FGMs are assumed to be temperature-dependent. The equations of motion are solved by a two step perturbation technique to determine the nonlinear frequencies of the FGM shells. Numerical results demonstrate that in most cases the natural frequencies of the FGM shells are increased but the nonlinear to linear frequency ratios of the FGM shells are decreased with increase in foundation stiffness. The results confirm that in most cases Voigt model and Mori-Tanaka model have the same accuracy for predicting the vibration characteristics of FGM shells.     

Benchmark solutions for functionally graded thick plates resting on Winkler–Pasternak elastic foundations

Exact solutions for functionally graded thick plates are presented based on the three-dimensional theory of elasticity. The plate is assumed isotropic at any point, while material properties to vary exponentially through the thickness. The system of governing partial differential equations is reduced to an ordinary one about the thickness coordinate by expanding the state variables into infinite dual series of trigonometric functions. Interactions between the Winkler–Pasternak elastic foundation and the plate are treated as boundary conditions. The problem is finally solved using the state space method. Effects of stiffness of the foundation, loading cases, and gradient index on mechanical responses of the plates are discussed. It is established that elastic foundations affects significantly the mechanical behavior of functionally graded thick plates. Numerical results presented in the paper can serve as benchmarks for future analyses of functionally graded thick plates on elastic foundations.     

Static and dynamic analysis of an FGM doubly curved panel resting on the Pasternak-type elastic foundation

The static, dynamic, and free vibration analysis of a functionally graded material (FGM) doubly curved panel are investigated analytically in the present paper. The FGM Panel is originated from a rectangular planform and its principle curvatures are considered to be constant. All mechanical properties of the FGM panel are assumed to vary continuously through the thickness according to a power law formulation except Poisson’s ratio, which is kept constant. A Pasternak-type elastic foundation containing damping effects is considered to be in contact with the panel during deformation. The elastic foundation reacts in both compression and tension. Equations of motion are established based on the first order shear deformation and the modified Sanders shell theories. Following the Navier type solution, the established equations are reduced to time-dependent ordinary differential equations. Using the Laplace transform, the time-dependency of the problem is eliminated. The solutions are obtained analytically in the Laplace domain and then are inverted to the time domain following an analytical procedure. Finally, the analytical results are verified with those reported in the literature.     

Vibrations of cracked rectangular FGM thick plates

Accurate first-of-its-kind solutions of the free vibration characteristics of side-cracked rectangular functionally graded material (FGM) thick plates are reported. From a brief review summary of available shear deformable plate theories, the well-established Reddy third-order plate theory apropos to cracked FGM thick plates is utilized. A novel Ritz procedure is developed incorporating special admissible functions – appropriately named in this study as crack functions – that properly account for the stress singularity behaviors in the neighborhood of a crack tip, and that properly account for the discontinuities of displacements and slops across a crack. Material properties of the FGM plates are assumed to vary continuously in the thickness direction according to the Mori–Tanaka scheme or a simple power law. The proposed special admissible functions accelerate the convergence of the extensive non-dimensional frequency solutions summarized. The first known non-dimensional frequencies of simply-supported and cantilevered cracked aluminum (Al) and ceramic (zirconia (ZrO₂)) or alumina (Al₂O₃) FGM thick plates of moderate thickness ratio (side-length to plate thickness, b/h = 10) are accurately determined. The effects of the volume fraction in the modeling of material distribution in the thickness direction and of cracks with different lengths, locations and orientations on the non-dimensional frequencies are investigated.     

Buckling and vibrations of two-directional FGM Mindlin circular plates under hydrostatic peripheral loading

An analysis is presented for the effect of hydrostatic peripheral loading on the free axisymmetric vibrations of functionally graded moderately thick circular plates on the basis of first-order shear deformation theory. The mechanical properties of the plate material are supposed to vary according to a power-law in both the radial and transverse directions. The numerical solution of the governing differential equation derived by using Hamilton's energy principle for such simply supported and clamped boundary conditions has been obtained employing the harmonic differential quadrature method choosing zeros of Chebyshev–Gauss–Lobatto as the grid points. The effect of different parameters has been analyzed on the frequency parameter for the first three modes of vibration. The critical buckling loads for both the plates have been computed by putting the frequencies to zero. Three-dimensional mode shapes for particular plate have been plotted. Obtained results have been compared.     

Wave propagation and transient response of a FGM plate under a point impact load based on higher-order shear deformation theory

In this paper, the wave propagation and transient response of an infinite functionally graded plate under a point impact load are presented. The effective material properties of functionally graded materials (FGMs) for the plate are assumed to vary continuously through the plate thickness and be distributed according to a volume fraction power law along the plate thickness. Based on the higher-order shear deformation theory and considering the effect of the rotary inertia, the governing equations of the wave propagation in the functionally graded plate are derived by using the Hamilton’s principle. The analytic dispersion relation of the functionally graded plate is obtained by means of integral transforms and a complete discussion of dispersion for the functionally graded plate is given. Then, using the dispersion relation and integral transforms, exact integral solutions for the functionally graded plate under a point impact load are obtained. The transient response curves of the functionally graded plates are plotted and the influence of volume fraction distributions on transient response of functionally graded plates is analyzed. Finally, the solutions of the higher-order shear deformation theory and the first-order shear deformation theory are studied.     

Nonlinear bending analysis of annular FGM plates using higher-order shear deformation plate theories

This paper addresses the axisymmetric nonlinear bending analysis of an annular functionally graded plate under mechanical loading based on FSDT and TSDT. Using nonlinear von-Karman theory, the discretized equations are solved using the dynamic relaxation (DR) method combined with the finite difference technique. The effects of the material constant n, boundary conditions, thickness-to-radius ratio and shear deformation are studied. The results show that although, the difference between TSDT and FSDT becomes greater with an increasing thickness-to-external radius ratio, the effects of different types of boundary conditions is also of great importance.     

Mechanical and thermal postbuckling of higher order shear deformable functionally graded plates on elastic foundations

This paper presents an analytical investigation on the buckling and postbuckling behaviors of thick functionally graded plates resting on elastic foundations and subjected to in-plane compressive, thermal and thermomechanical loads. Material properties are assumed to be temperature independent, and graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of constituents. The formulations are based on higher order shear deformation plate theory taking into account Von Karman nonlinearity, initial geometrical imperfection and Pasternak type elastic foundation. By applying Galerkin method, closed-form relations of buckling loads and postbuckling equilibrium paths for simply supported plates are determined. Analysis is carried out to show the effects of material and geometrical properties, in-plane boundary restraint, foundation stiffness and imperfection on the buckling and postbuckling loading capacity of the plates.     

Three-dimensional solution of axisymmetric bending of functionally graded circular plates

Based on three-dimensional theory, this paper investigates the axisymmetric bending of transversely isotropic and functionally graded circular plates subject to arbitrarily transverse loads using the direct displacement method. The material properties can arbitrarily vary along the thickness of the plate. The transverse load is expanded in the Fourier–Bessel series and superposition principle is then used to obtain the total response based on the results of each item of the series. For one item of the series of the load, we assume the distributions of the displacements in the radial direction and therefore only the distributions of the displacements in thickness direction are required to find. If the material properties vary in an exponential law, the exact solutions can be obtained for elastic simple support and rigid slipping support, which are satisfied on the every point of the boundaries. Moreover, the analytical solutions are also presented for simply supported and clamped conditions, which are satisfied using Saint Venant principle. Simultaneously, through the layerwise method a semi-analytical solution is proposed for the case of arbitrary variation of the material properties. Finally the numerical examples are presented to verify the proposed method.     

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 orthotropic rectangular plate resting on Pasternak elastic foundation under biaxial in-plane loading

In this study, buckling of rectangular orthotropic plates resting on a Pasternak elastic foundation under biaxial in-plane loading by the power series method (the method of Frobenius) was analyzed. Similar to many studies, two opposite edges of loading are simply supported and two other edges are assumed clamped. In order to extract the characteristic equations of orthotropic rectangular plate under in-plane loading resting on a Pasternak elastic foundation, the classical plate theory, by considering the interaction between plate and foundation, is used. The results showed that in the aspect ratio of less than 2, the existing Pasternak foundation caused the buckling load to increase severely, but by increasing the aspect ratio, the effect of the foundation is negligible. Applying the in-plane load in the y-direction caused the buckling load to decrease, but by increasing the aspect ratios the effect of the load is negligible.     

Nonlinear dynamic response and vibration of imperfect shear deformable functionally graded plates subjected to blast and thermal loads

Based on Reddy's higher-order shear deformation plate theory, this article presents an analysis of the nonlinear dynamic response and vibration of imperfect functionally graded material (FGM) thick plates subjected to blast and thermal loads resting on elastic foundations. The 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. Numerical results for the dynamic response and vibration of the FGM plates with two cases of boundary conditions are obtained by the Galerkin method and fourth-order Runge–Kutta method. The results show the effects of geometrical parameters, material properties, imperfections, temperature increment, elastic foundations, and boundary conditions on the nonlinear dynamic response and vibration of FGM plates.     

Accurate solution for free vibration analysis of functionally graded thick rectangular plates resting on elastic foundation

Vibration analysis of a functionally graded rectangular plate resting on two parameter elastic foundation is presented here. The displacement filed based on the third order shear deformation plate theory is used. By considering the in-plane displacement components of an arbitrary material point on the mid-plane of the plate and using Hamilton’s principle, the governing equations of motion are obtained which are five highly coupled partial differential equations. An analytical approach is employed to decouple these partial differential equations. The decoupled equations of functionally graded rectangular plate resting on elastic foundation are solved analytically for levy type of boundary conditions. The numerical results are presented and discussed for a wide range of plate and foundation parameters. The results show that the Pasternak (shear) elastic foundation drastically changes the natural frequency. It is also observed that in some boundary conditions, the in-plane displacements have significant effects on natural frequency of thick functionally graded plates and they cannot be ignored.     

Influences of elastic foundations and shear deformations on the buckling behavior of functionally graded material truncated conical shells under axial compression

This article presents an investigation on the buckling of functionally graded (FG) truncated conical shells under an axial load resting on elastic foundations within the shear deformation theory (SDT). The governing equations are solved using the Galerkin method, and the closed-form solution of the axial buckling load for FG conical shells on elastic foundations within the SDT is obtained. Various numerical examples are presented and discussed to verify the accuracy of the closed-form solution in predicting dimensionless buckling loads for FG conical shells on the Winkler–Pasternak elastic foundations within the SDT.     

Free vibration and flexural response of functionally graded plates resting on Winkler–Pasternak elastic foundations using nonpolynomial higher-order shear and normal deformation theory

In the present work, the flexural and vibration response of a functionally graded plate resting on Pasternak elastic foundation is analyzed using a recently developed nonpolynomial higher-order shear and normal deformation theory by the authors. The novelty of this theory is that it contains only four unknowns and also accommodates the thickness stretching effect. Two kinds of micromechanics models, namely, the Voigt and Mori–Tanaka models, are considered. Material properties of the functionally graded plates are assumed to vary continuously in the thickness direction according to either a simple power law or an exponential law. Finite element formulation is done using C° continuous Lagrangian quadrilateral nine-noded elements with eight degrees of freedom per node. The equations of motion are derived using a variational approach. Convergence and comparison studies are carried out to establish the authenticity and reliability of the solutions. The effect of various boundary conditions, geometric conditions, micromechanics models, and foundation parameters on the flexural and vibration response of the functionally graded plate are investigated.     

Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores

The bending response of functionally graded material (FGM) sandwich plates subjected to thermomechanical loads is investigated using a four-variable refined plate theory. A new type of FGM sandwich plate, namely, both FGM face sheets and an FGM hard core, is considered. Containing only four unknown functions, the governing equations are deduced based on the principle of virtual work and then these equations are solved via the Navier approach. Analytical solutions are obtained to predict the deflections and stresses of simply supported FGM sandwich plates. Benchmark comparisons of the solutions obtained for a degradation model (functionally graded face sheets and homogeneous cores) with ones computed by several other theories are conducted to verify the accuracy and efficiency of the present approach. The influences of volume fraction distribution, geometrical parameters, and thermal load on dimensionless deflections and normal and shear stresses of the FGM sandwich plates are studied.     

Exact solution for stability analysis of moderately thick functionally graded sector plates on elastic foundation

In this article, an exact analytical solution for buckling analysis of moderately thick functionally graded (FG) sector plates resting on Winkler elastic foundation is presented. The equilibrium equations are derived according to the first order shear deformation plate theory. Because of the coupling between the bending and stretching equilibrium equations of FG plates, these plates have deflection under in-plane loads lower than the critical buckling load acting on the mid-plane. The conditions under which FG plates remain flat in the pre-buckling configuration are investigated and the stability equations are obtained based on the flat plate assumption in the pre-buckling state. The stability equations are simplified into decoupled equations and solved analytically for plates having simply supported boundary condition on the straight edges. The critical buckling load is obtained and the effects of geometrical parameters and power law index on the stability of functionally graded sector plates are studded. The results for the critical buckling load of moderately thick functionally graded sector plates resting on elastic foundation are reported for the first time.     

Nonlinear aero-thermoelastic analysis of homogeneous and functionally graded plates in supersonic airflow using coupled models

In this paper, the aeroelastic behavior of homogeneous and functionally graded two- and three-dimensional flat plates is studied under supersonic airflow. The effects of coupled modeling of the aerodynamic heating with flight conditions and the thermal degradation of the plate are investigated, too. The von-Karman nonlinear strains, piston theory and a combination of simple rule of mixtures and the Mori–Tanaka scheme are used to model the structure, aerodynamic and material, respectively. The derived equations of motion are solved using Galerkin’s procedure along with the Runge–Kutta numerical technique. Finally, some examples are solved using the developed program and some conclusions are made. It is shown that under real flight conditions and using coupled model, the aerodynamic heating is very severe and the type of instability is divergence.