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.     

Thermal Buckling of Piezoelectric Functionally Graded Material Beams

Buckling analysis of functionally graded material (FGM) beams with surface-bonded piezoelectric layers which are subjected to both thermal loading and constant voltage is studied. The material nonhomogeneous properties are assumed to vary smoothly by distribution of power law through the beam thickness. The Euler-Bernoulli beam theory and nonlinear strain-displacement relation are used to obtain the governing equations of piezoelectric FGM beam. Beam is assumed under three types of thermal loading and various types of boundary conditions. For each case of thermal loading and boundary conditions, closed-form solutions are obtained. The effects of the applied actuator voltage, beam geometry, boundary conditions, and power law index of functionally graded material on the buckling temperature are investigated.     

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.     

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.     

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.     

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.     

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.     

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.     

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 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.     

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.     

GEOMETRICALLY NONLINEAR ANALYSIS OF TIMOSHENKO BEAMS UNDER THERMOMECHANICAL LOADINGS

Considering the axial extension and the transversal shear deformation, geometrically nonlinear governing equations for static deformations of Timoshenko beams subjected to thermal as well as mechanical loadings are formulated. As an example, on the basis of the governing equations, thermal postbuckling response of an immovably pinned-fixed Timoshenko beam subjected to a static transversely nonuniform temperature rise is numerically analyzed by using a shooting method. Characteristic curves showing the relationships between the beam deformation and temperature rise are presented. The thermal postbuckled configurations and the equilibrium paths of the beam are presented. In particular, the effects of shear deformation on the buckling response are quantitatively investigated. The numerical results show, as we know, that shear deformation effects become significant with decrease of the slenderness and with increase of the shear flexibility.     

Hygrothermal buckling analysis of magnetically actuated embedded higher order functionally graded nanoscale beams considering the neutral surface position

In this article, the effects of humidity and thermal loads on buckling behavior of functionally graded (FG) nanobeams resting on elastic foundation and subjected to a unidirectional magnetic field is investigated. The nanobeam is modeled using different higher order refined beam theories which capture shear deformation influences needless of shear correction factors. The neutral axis position for all proposed beam models is determined. The material properties of FG nanobeam are temperature dependent and change gradually in spatial coordinate through the sigmoid and power-law models. Small-scale behavior of the nanobeam is described applying nonlocal elasticity theory of Eringen. Nonlocal governing equations for an embedded nanosize functionally graded material beam under hygrothermal loads obtained from Hamilton's principle are solved by an analytic method which satisfies various boundary conditions including S–S, C–S, and C–C. The validation of developed refined beam model has been proved with comparison to a previously published work on FG nanobeams. Numerical results are calculated for various beam theories to reveal the influences of moisture and temperature rise, elastic medium, nonlocality, volume fraction index, boundary conditions, and longitudinal magnetic field on the hygrothermal buckling responses of nanoscale P-FGM and S-FGM beams. The present study would be useful in the design of the nanoscale systems as one of the most demanded technologies in the near future.     

Thermal Postbuckling Analysis of Functionally Graded Beams

Thermal buckling and postbuckling analysis of functionally graded (FG) beams is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. For discretizing the governing equations and the related boundary conditions differential quadrature method (DQM) as a simple and computationally efficient numerical tool is used. Based on displacement control method, a direct iterative method is employed to obtain thermal postbuckling behavior of FG beams with different boundary conditions and geometrical parameters.     

Thermal post-buckling of temperature dependent sandwich plates with FG-CNTRC face sheets

Present research investigates the thermal postbuckling of sandwich plates containing a stiff core and two thin carbon nanotube reinforced composite (CNTRC) face sheets. Properties of the core, carbon nanotubes (CNTs) and polymeric matrix of the faces are assumed to be temperature-dependent. It is assumed that CNTs as reinforcements may be distributed according to a functionally graded pattern. Plate is formulated based on the first-order shear deformation theory and von Kármán type of geometrical nonlinearity. The governing equations are obtained by the energy method with the aid of the Conventional Ritz method. Shape functions of the Ritz method are estimated according to the Chebyshev polynomials. A set of nonlinear eigenvalue equations is achieved. The obtained equations are homogeneous, coupled, and nonlinear in terms of both displacements and temperature. A successive displacement control strategy is implemented to trace the thermal postbuckling equilibrium path of the plate. It is shown that, with increasing the volume fraction of CNT, critical buckling temperature of sandwich plate increases and postbuckling deflection decreases. Furthermore, through a functionally graded distribution of volume fraction of CNTs across the thickness, critical buckling temperature of the sandwich plate may be enhanced and thermal postbuckling deflection may be alleviated.     

THERMAL BUCKLING AND POSTBUCKLING OF SYMMETRICALLY LAMINATED COMPOSITE PLATES

The thermal buckling and postbuckling response of symmetrically laminated composite plates are discussed. Using variational methods in conjunction with a Ray-leigh-Ritz formulation, thermal buckling and postbuckling are investigated for two laminates, a ( ±45/0/90) _s and a ( ± 45/0₂ ) _s, under two different simple support conditions, fixed and sliding. These laminates are subjected to the condition of a uniform temperature change. The effects of the principal material axes not being aligned with the edges of the plate, referred to here as material axis skewing, are also investigated. Although differences between buckling temperatures for the two support conditions were small, support conditions can have a large influence on thermal postbuckling response. In general, plates with fixed simple supports defied more than plates with sliding simple supports. In addition, support conditions can influence modal interaction. Skewing of the material axis decreases the buckling temperatures of both laminates and, like fixed support conditions, causes increased postbuckling deflections. Skewing also influences modal interaction.     

Postbuckling analysis of shear deformable FG shallow spherical shell panel under nonuniform thermal environment

In this article, the buckling and postbuckling behavior of functionally graded spherical shell panel is examined under nonuniform thermal environment. The effective material properties of the graded structure are evaluated using the Voigt's micromechanical model through the power-law distribution. For the analysis purpose, a general nonlinear higher order mathematical model is developed in conjunction with Green–Lagrange geometrical nonlinearity. The governing equation is derived using variational principle and solved through the direct iterative method. The effect of different geometrical and material parameters on the buckling and postbuckling responses of the functionally graded shell panels is examined and discussed in detail.     

On a Contact Problem of Two-Layer Beams Coupled by Boundary Conditions in a Temperature Field

We consider a mathematical model of two-layer beams coupled by boundary conditions in a stationary temperature field taking into account geometric nonlinearity. The stationary temperature field is defined by a 2D heat transfer equation with boundary conditions of the first kind. The geometric nonlinearity is introduced via von Kármán's relations for both beams. Equations of beam deflection are derived due to the Euler–Bernoulli hypothesis. The contact interaction is described using Winkler's model. Scenarios of a transition from regular to chaotic regimes are studied. Phase synchronization of beam vibrations versus both character and intensity of the applied temperature field is investigated.     

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.