Thermomechanical nonlinear analysis of axially compressed carbon nanotube-reinforced composite cylindrical panels resting on elastic foundations with tangentially restrained edges

Buckling, postbuckling, and nonlinear responses of composite cylindrical panels reinforced by single-walled carbon nanotubes (CNTs), supported by an elastic foundation, exposed to elevated temperature and axially compressed by uniform load are investigated in this article. Distribution of CNTs is uniform or graded in the thickness direction and the effective properties of CNT-reinforced composite are assumed to be temperature dependent, and are estimated by extended rule of mixture through a micromechanical model. Governing equations are established based on thin shell theory taking von Kármán–Donnell nonlinearity, initial geometrical imperfection, Pasternak-type elastic foundation and tangential elastic constraints of boundary edges into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and Galerkin method is applied to derive explicit expressions of load–deflection relation from which critical buckling loads can be obtained. Unlike works in the literature, the present study accounts for elasticity of tangential restraint of two unloaded straight edges in model of cylindrical panel. The study also gives conditions for which bifurcation type buckling response can occur and novel findings in numerical examples.     

Thermomechanical postbuckling behavior of CNT-reinforced composite sandwich plate models resting on elastic foundations with elastically restrained unloaded edges

Buckling and postbuckling behaviors of two models of sandwich plate reinforced by carbon nanotubes (CNTs) resting on elastic foundations and subjected to uniaxial compressive and thermomechanical loads are investigated in this paper. Material properties of all constituents are assumed to be temperature dependent and effective properties of CNT-reinforced composite layer are determined according to extended rule of mixture. Governing equations are established within the framework of first-order shear deformation theory taking into account von Kármán nonlinearity, initial geometrical imperfection, plate-foundation interaction and tangential elastic constraints of unloaded edges. Three types of loading are considered including uniaxial compression, preexisting thermal load combined with uniaxial compression and preexisting mechanical load combined with thermal load. Approximate analytical solutions are assumed to satisfy simply supported boundary conditions and the Galerkin method is used to derive nonlinear load-deflection relations from which buckling loads and postbuckling equilibrium paths are determined. The most important findings are that tangential constraints of unloaded edges significantly lowers buckling loads and postbuckling load capacity of sandwich plates and, in contrast, buckling loads and postbuckling strength are considerably enhanced as sandwich plate is constructed from CNT-reinforced composite core layer and homogeneous face sheets.     

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.     

Thermal buckling of temperature-dependent FG-CNT-reinforced composite skew plates

Critical buckling temperatures of skew plates made from a polymeric matrix reinforced by single-walled carbon nanotubes (CNTs) are obtained in the present research. Reinforcements are distributed across the thickness of the plate uniformly or according to a prescribed nonuniform function. All of the thermomechanical properties are assumed to be temperature dependent. First-order shear deformation plate theory is used as the basic assumption to obtain the total strain and potential energies of the plate due to the thermally induced prebuckling loads. A transformation is proposed to express the components of the displacement field in an oblique coordinate system. A Ritz-based solution is implemented to obtain the matrix representation of the stability equations associated to the onset of buckling. Gram–Schmidt process is used to obtain a set of orthogonal shape functions as the basis polynomials of the Ritz method. The obtained eigenvalue problem is solved successively to extract the critical buckling temperature of the skew plate. Convergence and comparison studies are provided to assure the accuracy and correctness of the proposed formulation. Afterward, parametric studies are given to explore the influences of boundary conditions, CNT volume fraction, CNT dispersion profile, aspect ratio, skew angle, and side to thickness ratio.     

Mechanical and thermal buckling of exponentially graded sandwich plates

This article presents an analytical solution for the mechanical and thermal buckling of exponentially graded material (EGM) sandwich plates. The solution is obtained using a four-variable refined plate theory. Two types of sandwich plates are investigated: one with EGM face sheets and homogeneous core; the other with EGM core and homogeneous face sheets. The governing equations are derived based on the principle of virtual work and then solved through Navier method. The results on critical buckling load and temperature change of simply supported EGM sandwich plates are obtained. The influences of several parameters on buckling behaviors are discussed.     

Size dependent thermal buckling and postbuckling of functionally graded circular microplates based on modified couple stress theory

In this article, size-dependent thermal buckling and postbuckling behavior of a functionally graded circular microplate under uniform temperature rise field and clamped boundary conditions is investigated. Material properties are assumed to gradually vary through the thickness according to a simple power law. Equilibrium equations and associated boundary conditions are derived using variational method and based on modified couple stress theory, classical plate theory and von Kármán geometric nonlinearity. The differential quadrature method is used to discretize the governing equations. This technique is accompanied by an iterative method to determine the thermal postbuckling behavior of microplate. Finally, effects of length scale parameter, power law index and ratio of thickness to radius on the thermal buckling and postbuckling behavior of FG circular microplate are investigated.     

Thermal buckling analysis of functionally graded perforated annular sector plates using 3D elasticity theory

Thermal buckling of annular sector plates with circular cutouts made of functionally graded material is analyzed in this article. Graded material is considered with two parts of ceramic structure made of zirconia and metal structure which is aluminum. Unlike most conducted research, the direction of property change is observed in three main directions. Thermal loading is assumed as a uniform increase in temperature influencing the whole sector. 3D-finite element method based on elasticity theory is used in this analysis, which resets first and second variations of potential energy of the sector to zero to find equilibrium and stability equations, respectively. Green’s nonlinear strain–displacement relations are used to obtain geometrical stiffness matrix. In the finite element method, unlike most studies, a 3D eight-nodded element is used, which has nodes in the direction of thickness. The circular cutouts of the sector have added to the complexity of the analysis. The finite element formulation is coded in MATLAB. Finally, the effect of different parameters such as dimension and number of cutouts, power law index, and orientation of graded material on the critical thermal buckling temperature is studied.     

Thermal buckling and postbuckling analysis of piezoelectric FGM beams based on high-order shear deformation theory

This article deals with the thermal buckling and postbuckling of functionally graded material (FGM) beams with surface-bonded piezoelectric actuators based on physical neutral surface concept and high-order shear deformation theory including von Kármán strain–displacement relationships. The beams are exposed to a uniform temperature field and electric field, the material properties of FGM layers are temperature-dependent and vary in the thickness direction. The approximate solutions of piezoelectric FGM beams for thermal buckling and postbuckling are obtained by a two-step perturbation method, meanwhile, the analytical solutions of Timoshenko beam model and Euler beam model are also presented. The validity of the present work can be confirmed by comparisons with previous results. The effects of the applied actuator voltage, beam geometry as well as volume fraction index of FGM beam on the critical buckling temperature, and postbuckling load–deflection relationships are investigated.     

Thermal buckling and post-buckling of FGM Timoshenko beams on nonlinear elastic foundation

Buckling and post-buckling thermomechanical deformations of a functionally graded material (FGM) Timoshenko beam resting on a two-parameter non-linear elastic foundation and subjected to only a temperature rise have been numerically investigated with the shooting method. The material properties are assumed to vary only in the thickness direction according to a power law function. Through-the-thickness temperature distribution is determined by numerically solving the one-dimensional heat conduction equation. Geometric non-linearities in the strain-displacement relations and the non-linear traction-displacement relations at the interface between the beam and the foundation are considered. For clamped-clamped and immovable simply supported beams, critical values of the ratio of temperatures of the top and the bottom surfaces of the beam for transitions in buckling modes to occur are determined. Post-buckled equilibrium paths and configurations of the heated FGM beam are illustrated for different values of the elastic foundation stiffness parameters, exponent in the power law variation of material properties and the slenderness ratio. Results for the Timoshenko beam are compared with those of the corresponding homogeneous Euler–Bernoulli beam available in the literature.     

Thermal buckling of temperature-dependent composite super elliptical plates reinforced with carbon nanotubes

In this work, the problem of thermal buckling of composite plates reinforced with carbon nanotubes (CNTs) is investigated. Distribution of CNTs as reinforcements through the thickness direction of the plate is assumed to be either uniform or functionally graded (FG). Properties of the reinforcement and matrix are both temperature dependent. Properties of the composite media are obtained according to a refined rule of mixture approach where the e?ciency parameters are introduced. The plate is in a super elliptical shape where the simple elliptical shape and rectangular shapes are obtained as especial cases. In these types of plates due to the round corners, stress concentration phenomenon is eliminated. Based on the Ritz method where the shape functions are of the polynomial type, the governing equations are obtained. These equations are solved using an iterative eigenvalue problem since the properties are temperature dependent. Numerical results are validated for the simple case of an isotropic plate. Novel numerical results are provided for plates reinforced with CNTs in different shapes, various volume fractions and different patterns of CNT distribution. It is shown that FG-X pattern of CNTs in matrix results in the maximum critical buckling temperature.     

Pore pressure and porosity effects on bending and thermal postbuckling behavior of FG saturated porous circular plates

Postbuckling and nonlinear bending of a geometrically perfect circular plate arisen by the temperature of one side of a axisymmetric circular plate made from a poroelastic solid whose matrix pores have been saturated by fluid have been numerically analyzed. The plate porosity varies continuously through the plate thickness according to some given specific functions. The postbuckling and nonlinear bending configurations of respectively clamped and simply-supported plates, as well as the critical postbuckling temperature, have been obtained under the influence of the plate one side surface temperature, thickness, type of pore distribution, porosity and compressibility of fluid confined by the pores. Equilibrium equations of the plate have been derived on the basis of the classical plate theory, Love–Kirchhoff hypothesis and the Sander’s nonlinear strain–displacement relationship. They have been discretized via differential quadrature method. The numerical results appropriately coincide with the relevant literature, namely the results of saturated porous and mathematically equivalent plates made from functionally grated materials.     

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.     

Finite element analysis of hygrothermal effects of angle-ply composite plates resting on elastic foundation

Analytical and numerical results on hygrothermal behaviors of angle-ply composite plates resting on elastic foundation are scarce in literature. To fill this gap, a higher-order global-local model (HGLMA) is proposed for hygrothermal analysis of angle-ply composite plates resting on elastic foundation. Based on the model HGLMA, a triangular plate element is formulated by using the refined element approach, in which a new finite element formulation including effects of elastic foundation has been developed. By analyzing hygrothermal response of composite plates with general configurations, the effects of elastic foundation parameters, lamination angles, boundary conditions, transverse normal strain, and hygrothermal expansion coefficients on the hygrothermal response have all been studied. Numerical results showed that elastic foundation has a significant impact on the behaviors of composite plates and models neglecting transverse normal strain fail to simulate the effects of the elastic foundation parameters on displacements and stresses of symmetric composite plates.     

Effects of nonlinear hygrothermomechanical loading on bending of FGM rectangular plates resting on two-parameter elastic foundation using four-unknown plate theory

In the present study, a simple four-unknown exponential shear deformation theory is developed for the bending of functionally graded material (FGM) rectangular plates resting on two-parameter elastic foundation and subjected to nonlinear hygrothermomechanical loading. The elastic properties, coefficient of thermal expansion, and coefficient of moisture expansion of the plate are assumed to be graded in the thickness direction according to a simple power-law distribution in terms of volume fractions of material constituents. Unlike first-order and other higher-order plate theories, the present theory has four independent unknowns. The in-plane displacement field of the present theory uses exponential functions in terms of thickness co-ordinate for calculating out-of-plane shearing strains. The transverse displacement includes bending and shear components. The principle of virtual displacement is employed to derive the governing equations and associated boundary conditions. A Navier solution technique is employed to obtained an analytical solutions. The elastic foundation is modelled as two-parameter Winkler–Pasternak foundation. The numerical results obtained are compared with previously published results wherever possible to prove the efficacy and accuracy of the present theory. The effects of stiffness and gradient index of the foundation on the hygrothermomechanical responses of the plates are discussed.     

Thermal buckling and postbuckling responses of geometrically imperfect FG porous beams based on physical neutral plane

Abstract

Considering the third-order shear deformation and physical neutral plane theories, thermal postbuckling analysis for functionally graded (FG) porous beam are performed in this research. The cases of shear deformable functionally graded materials (FGM) beams with initial deflection and uniformly distributed porosity are considered. Geometrically imperfect FG porous beams with two different types of immovable boundary conditions as clamped–rolling and clamped–clamped are analyzed. Thermomechanical nonhomogeneous material properties of the FG porous beam are assumed to be temperature and position dependent. FG porous beams are subjected to different types of thermal loads as heat conduction and uniform temperature rise. Heat conduction equation is solved analytically using the polynomial series solution for the one-dimensional condition. The governing equilibrium equations are obtained by applying the virtual displacement principle. Assuming von Kármán type of geometrical nonlinearity, equilibrium equations are nonlinear and are solved using an analytical method. A two-step perturbation technique is used to obtain the thermal buckling and postbuckling responses of FG porous beams. The numerical results are compared with the case of perfect FGM Timoshenko beams without porosity distribution based on the midplane formulation. Parametric studies of the perfect/imperfect FG porous beams for two types of thermal loading and boundary conditions are provided.     

Nonlinear vibration and dynamic response of shear deformable imperfect functionally graded double-curved shallow shells resting on elastic foundations in thermal environments

In this article, nonlinear vibration and dynamic response of imperfect functionally graded materials (FGM) thick double-curved shallow shells resting on elastic foundations are investigated using Reddy's third-order shear deformation shell theory in 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 the constituents. The FGM shells are subjected to mechanical, damping, and thermal loads. The Galerkin method and fourth-order \hboxRunge–Kutta method are used to calculate natural frequencies, nonlinear frequency–amplitude relation, and dynamic response of the shells. In numerical results, the effects of geometrical parameters, material properties, imperfections, shear deformation, the elastic foundations, mechanical, thermal and damping loads on the nonlinear dynamic response, and nonlinear vibration of FGM double-curved shallow shells are investigated. Accuracy of the present formulation is shown by comparing the results of numerical examples with the ones available in literature.     

Thermal buckling analysis of FGM sandwich truncated conical shells reinforced by FGM stiffeners resting on elastic foundations using FSDT

This work presents an analytical approach to investigate the mechanical and thermal buckling of functionally graded materials sandwich truncated conical shells resting on Pasternak elastic foundations, subjected to thermal load and axial compressive load. Shells are reinforced by closely spaced stringers and rings, in which the material properties of shells and stiffeners are graded in the thickness direction following a general sigmoid law distribution and a general power law distribution. Four models of coated shell-stiffener arrangements are investigated. The change of spacing between stringers in the meridional direction also is taken into account. Two cases on uniform temperature rise and linear temperature distribution through the thickness of shell are considered. Using the first-order shear deformation theory, Lekhnitskii smeared stiffener technique and the adjacent equilibrium criterion, the linearization stability equations have been established. Approximate solution satisfies simply supported boundary conditions and Galerkin method is applied to obtain closed-form expression for determining the critical compression buckling load and thermal buckling load in cases uniform temperature rise and linear temperature distribution across the shell thickness. The effects of temperature, foundation, core layer, coating layer, stiffeners, material properties, dimensional parameters and semi-vertex angle on buckling behaviors of shell are shown.     

Thermal buckling and vibration of functionally graded sinusoidal microbeams incorporating nonlinear temperature distribution using DQM

Thermal buckling and vibration of functionally graded (FG) sinusoidal microbeams with temperature-dependent properties and three kinds of temperature distributions are investigated in this article. As one material length scale is introduced, the modified couple stress theory is capable of predicting the small-scale effects. Material properties of FG microbeams are calculated using the Mori–Tanaka method. Furthermore, temperature-dependent properties are taken into account to investigate the mechanical characteristics of FG microbeams in high–thermal-gradient environment. Motion equations and the associated boundary conditions are obtained simultaneously through variational principle. Then Navier procedure and the differential quadrature method incorporating an iterative procedure are used to solve the governing differential equations with temperature-dependent properties and general boundary conditions. Numerical examples are performed for demonstrating the influences of temperature distribution, beam thickness, material length scale, slenderness ratio, shear deformation, functionally graded index, boundary conditions, and temperature-dependent/independent properties on thermal buckling and free vibration behaviors of FG microbeams.     

The buckling of FGM truncated conical shells subjected to axial compressive load and resting on Winkler–Pasternak foundations

In this study, the buckling analysis of the simply supported truncated conical shell made of functionally graded materials (FGMs) is presented. The FGM truncated conical shell subjected to an axial compressive load and resting on Winkler–Pasternak type elastic foundations. The material properties of functionally graded shells are assumed to vary continuously through the thickness. The modified Donnell type stability and compatibility equations are solved by Galerkin’s method and the critical axial load of FGM truncated conical shells with and without elastic foundations have been found analytically. The appropriate formulas for homogenous and FGM cylindrical shells with and without elastic foundations are found as a special case. Several examples are presented to show the accuracy and efficiency of the formulation. Finally, parametric studies on the buckling of FGM truncated conical and cylindrical shells on elastic foundations are being investigated. These parameters include; power-law and exponential distributions of FGM, Winkler foundation modulus, Pasternak foundation modulus and aspect ratios of shells.     

Stochastic analysis of laminated composite plates on elastic foundation: The cases of post-buckling behavior and nonlinear free vibration

This study deals with the stochastic post-buckling and nonlinear free vibration analysis of a laminated composite plate resting on a two parameters Pasternak foundation with Winkler cubic nonlinearity having uncertain system properties. The system properties are modeled as basic random variables. A C⁰ nonlinear finite element formulation of the random problem based on higher-order shear deformation theory in the von Karman sense is presented. A direct iterative method in conjunction with a stochastic nonlinear finite element method proposed earlier by the authors is extended to analyze the effect of uncertainty in system properties on the post-buckling and nonlinear free vibration of the composite plates having Winler type of geometric nonlinearity. Mean as well as standard deviation of the responses have been obtained for various combinations of geometric parameters, foundation parameters, stacking sequences and boundary conditions and compared with those available in the literature and Monte Carlo simulation.