Thermal postbuckling of temperature-dependent sandwich beams with carbon nanotube-reinforced face sheets

Thermal postbuckling response of a sandwich beam made of a stiff host core and carbon nanotube (CNT)-reinforced face sheets is analyzed in this research. Distribution of CNTs across the thickness of face sheets may be uniform or functionally graded. Material properties of the constituents are considered as temperature dependent. Properties of the face sheets are obtained by means of a modified rule of mixture approach. First-order shear deformation theory and von Kármán type of geometrical nonlinearity are incorporated with the virtual displacement principle. Ritz method with polynomial basis functions is applied to the virtual displacement principle to obtain the matrix representation of the governing equations. An iterative displacement control algorithm is applied to solve the nonlinear eigenvalue problem and trace the postbuckling equilibrium path. It is shown that, graded profile of CNTs, length to thickness ratio, host thickness to face thickness ratio, volume fraction of CNTs, boundary conditions, and temperature dependency, all are important factors on critical buckling temperature and postbuckling equilibrium path of sandwich beams with CNT-reinforced face sheets. However, influence of host thickness to face thickness ratio is ignorable.     

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 and postbuckling behavior of functionally graded carbon-nanotube-reinforced composite plates resting on elastic foundations with tangential-edge restraints

This article presents an analytical approach to investigate the buckling and postbuckling behavior of functionally graded nanocomposite plates reinforced by single-walled carbon nanotubes, resting on elastic foundations and subjected to thermal load due to uniform temperature rise or linear temperature change across the plate thickness. The material properties of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) are assumed to be temperature independent, graded in the thickness direction, and estimated by extended rule of mixture through a micromechanical model. Formulations are based on classical plate theory taking von Kármán nonlinearity, initial geometrical imperfection, Pasternak-type foundation interaction, and tangential-edge constraints into consideration. Approximate solutions of deflection and stress functions are assumed to satisfy simply supported boundary conditions, and the Galerkin method is applied to obtain closed-form expressions of buckling temperatures and temperature-deflection relations. The influences of carbon-nanotube volume fraction and distribution pattern, aspect ratios, stiffness of foundations, degree of tangential-edge constraints, and imperfection on the thermal buckling and postbuckling behavior of FG-CNTRC plates are analyzed and discussed.     

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.     

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 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.     

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.     

Thermal buckling analysis of moderately thick FGM plates based on the von Kármán nonlinearity and improved third order shear deformation theory

Abstract

In this study, thermal buckling of moderately thick functionally graded rectangular plates with all edges simply supported is analyzed by means of an improved third order shear deformation theory (improved TSDT). The plate is assumed to be under two types of thermal loadings, namely; uniform temperature rise and nonlinear temperature change across the thickness. The equilibrium and stability equations are derived based on the von Kármán type of geometrical nonlinearity and the improved third-order theory. By solving the stability equations, the value of buckling temperature difference is obtained. To calculate the critical buckling temperature difference, this value is minimized with respect to the half-wave parameters. The results are compared with the known data in literatures. The results indicate that, the values of critical buckling temperature difference which are obtained based on the improved TSDT, are lower in comparison with those obtained based on TSDT. Also, the results show that incorporation of the von Kármán type of geometrical nonlinearity with the improved third-order theory, gives the lower values of the critical buckling temperature difference.     

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.     

Effect of Initial Imperfections on Thermal Buckling of Functionally Graded Plates

ABSTRACT

Thermal buckling analysis of rectangular functionally graded plates with initial geometrical imperfections is presented in this article. The equilibrium, stability, and compatibility equations of an imperfect functionally graded plate are derived using the first-order shear deformation plate theory. It is assumed that the nonhomogeneous mechanical properties of the plate, graded through the thickness, are described by a power function of the thickness variable. The plate is assumed to be under three types of thermal loading, namely: uniform temperature rise, nonlinear temperature rise through the thickness, and axial temperature rise. Resulting equations are employed to obtain the closed-form solutions for the critical buckling temperature change of an imperfect functionally graded plate. The influence of transverse shear on thermal buckling load is discussed.     

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.     

Thermal Stability of FGM Sandwich Plates Under Various Through-the-Thickness Temperature Distributions

A thermal stability analysis of functionally graded material (FGM) isotropic and sandwich plates is carried out by virtue of a refined quasi-3D Equivalent Single Layer (ESL) and Zig-Zag (ZZ) plate models developed within the framework of the Carrera Unified Formulation (CUF) and implemented within the Hierarchical Trigonometric Ritz Formulation (HTRF). The Principle of Virtual Displacements (PVD) is used both to derive the thermal stability differential equations with natural boundary conditions and to develop the HTRF. Uniform, linear, and non-linear temperature rises through-the-thickness direction are taken into account. The non-linear temperature distribution is given in different forms: 1) functionally graded; 2) solution of the one-dimensional Fourier heat conduction equation; and 3) sinusoidal. Several FGM sandwich plate configurations are investigated. Parametric studies are carried out in order to evaluate the effects of significant parameters, such as volume fraction index, length-to-thickness ratio, boundary conditions, aspect ratio, sandwich plate type, and temperature distribution through-the-thickness direction, on the critical buckling temperatures.     

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.     

Axisymmetric nonlinear rapid heating of FGM cylindrical shells

Large amplitude thermally induced vibrations of cylindrical shells made of a through-the-thickness functionally graded material (FGM) are investigated in the current research. All of the thermo-mechanical properties of the FGM shell are assumed to be functions of temperature and thickness coordinate. Shell is subjected to rapid surface heating on the ceramic-rich surface while the other surface of the shell is kept at reference temperature. One dimensional heat conduction equation is constructed and solved by means of a hybrid finite difference-Crank–Nicolson algorithm. The constructed heat conduction equation is nonlinear since the thermal conductivity is temperature dependent. With the aid of first-order shear deformation shell theory under the axisymmetric Donnell kinematic assumptions and von Kármán type of strain-displacement relations, the total energy of the shell is established. Implementing the conventional Ritz method, a set of nonlinear coupled algebraic equations are obtained which govern the dynamics of the shell under thermal shock. These equations are solved in time domain using the Newmark time marching scheme and the simple Picard successive method. Parametric studies are given to explore the dynamics of an FGM cylindrical shell under thermal shock.     

Dynamic Thermal Postbuckling Analysis of Piezoelectric Functionally Graded Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is analyzed using an incremental numerical technique. The shell is graded across the thickness according to a power law form function. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the classical shell theory with Sanders' nonlinear kinematic relations. Then, using Hamilton's principle, equations of motion are derived for the piezoelectric FGM cylindrical shell. A finite difference based method combined with the Runge–Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent of FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known data in the literature.     

Geometrically Non-Linear Rapid Heating of Temperature-Dependent Circular FGM Plates

Based on the uncoupled thermoelasticity assumptions, axisymmetric thermally induced vibrations of a circular plate made of functionally graded materials (FGMs) are analyzed. Each thermomechanical property of the circular plate is assumed to be functions of temperature and thickness coordinate. Solution of the transient one-dimensional heat conduction equation with the arbitrary type of time-dependent boundary conditions is carried out employing the central finite difference method combined with the Crank–Nicolson time marching scheme. Afterwards, with the establishment of the associated Hamilton's principle and the accountancy of the von Kármán type of geometrical non-linearity, the motion equations are obtained with the aid of the conventional multi-term Ritz method. The solution of highly coupled non-linear motion equations is obtained utilizing a hybrid iterative Newton–Raphson–Newmark scheme. After validating the developed computer code, some parametric studies are accomplished to show the influences of various involved parameters. It is shown that temperature dependency, geometrical non-linearity, plate thickness, power law index, and the type of thermal in-plane and out-of-plane mechanical boundary conditions, all affect the temporal evolution of plate characteristics.     

Elastoplastic analysis of FGM plate with a central cutout of various shapes under thermomechanical loading

The present work aims to study the elastoplastic buckling, postbuckling, and failure behavior of perforated Ni/Al₂O₃ functionally graded material (FGM) plate with various shaped cutouts (i.e., circular, square, diamond, and elliptical) of various sizes under thermomechanical loading conditions using finite element method (FEM). The nonlinear FEM formulation is based on the first-order shear deformation theory and von Kármán’s nonlinear kinematics in which the material nonlinearity is incorporated. The nonlinear temperature-dependent thermoelastic material properties of FGM plate are varied in the thickness direction by controlling the volume fraction of the constituent materials (i.e., ceramic and metal) as per a power law, and Mori–Tanaka homogenization scheme is applied to evaluate the properties at a particular thickness coordinate of FGM. In accordance with the Tamura–Tomota–Ozawa model (TTO model), the ceramic phase of FGM is considered to be elastic, whereas the metal phase is assumed to be elastoplastic. Further, the elastoplastic analysis of FGM is assumed to follow J₂ plasticity with isotropic hardening. After validating the present formulation with the results available in the literature, various numerical studies are conducted to examine the effects of material inhomogeneity, thermal loading, cutout shape, and size on the elastoplastic buckling, postbuckling, and failure behavior of perforated FGM plate. It is observed that for smaller cutout sizes, the FGM plate with square shape cutout possesses maximum value of ultimate failure load; however, for larger cutout size, the FGM plate with diamond cutout depicts highest ultimate failure load. Furthermore, for all cutout shapes, the ultimate failure load of FGM plate decreases with an increase in cutout size. It is also revealed that irrespective of shape and size of cutout, the material plastic flow has considerable effect on postbuckling path of FGM plate, and under thermomechanical loading conditions, the FGM plate shows destabilizing response after the point of maximum postbuckling strength.     

Dynamic Thermal Postbuckling Analysis of Shear Deformable Piezoelectric-FGM Cylindrical Shells

Dynamic thermal postbuckling behavior of functionally graded cylindrical shells with surface-bonded piezoelectric actuators subjected to the combined action of thermal load and applied actuator voltage is studied. The shell material is graded across the thickness according to a power law. The material properties of the functionally graded cylindrical shells are considered to be temperature dependent. The theoretical formulations are based on the Sanders nonlinear kinematic relations, which account for the transverse shear strains, and the third-order shear deformation shell theory is employed. Hamilton's principle is used to derive the equations of motion governing piezoelectric FGM cylindrical shells. A finite difference approximation combined with the Runge-Kutta method is employed to predict the postbuckling equilibrium paths, and the dynamic buckling temperature difference is detected according to Budiansky's stability criterion. Numerical results are presented to demonstrate the effects of the applied actuator voltage, shell geometry, volume fraction exponent in the power-law variation of the FGM, and the temperature dependency of the material properties on the postbuckling behavior of the shell. The results for simpler states are validated with the known results in the literature.     

Generalized differential quadrature nonlinear buckling analysis of smart SMA/FG laminated beam resting on nonlinear elastic medium under thermal loading

The nonlinear thermal buckling analysis of functionally graded (FG) beam integrated with shape memory alloy (SMA) layer(s), with different lay-up configurations and supported on a nonlinear elastic foundation, has been investigated. The FG layer is graded through the beam thickness direction and thermomechanical properties are assumed to be temperature dependent. The Brinson one-dimensional constitutive law are used to model the characteristics of SMA. The von Kármán strain–displacement fields with the Timoshenko beam theory are applied to the Hamilton’s principle to derive the set of nonlinear equilibrium equations. Generalized differential quadrature method along with direct iterative scheme is utilized to discretize and solve the nonlinear equilibrium equations. The accuracy of proposed model is compared and validated with previous research in literature. The detailed parametric study has been performed to investigate the influence of geometrical, material, and some other key parameters on the nonlinear thermal buckling solutions. The results show that selecting the proper lay-up is of great importance because the type of SMA/FG lay-up can considerably affect the nonlinear buckling solutions. Moreover, adequate application of SMA layers in a proper lay-up configuration significantly postpones the thermal buckling temperature of the beam.     

Thermoelastic Buckling Analysis of Functionally Graded Circular Plates Integrated with Piezoelectric Layers

Thermal buckling of circular plates made of functionally graded materials with surface-bounded piezoelectric layers are studied. The material properties of the FG plates are assumed to vary continuously through the plate thickness by distribution of power law of the volume fraction of the constituent materials. The general thermoelastic nonlinear equilibrium and linear stability equations for the piezoelectric FG plate are derived using the variational formulations. Buckling temperatures are derived for solid circular plates under uniform temperature rise, nonlinear and linear temperature variation through the thickness for immovable clamped edge of boundary conditions. The effects of piezo-to-host thickness ratio, applied actuator voltage, boundary condition, and power law index of functionally graded plates on the buckling temperature of plate are investigated. The results are verified with the data in literature.