Thermal Postbuckling Response Research Articles

Nonlinear thermal post-buckling analysis of rectangular FG plates using CUF

In this study, a numerical analysis examines the thermal post-buckling response of rectangular functionally graded (FG) plates. Since the post-buckling is analyzed in a thermal environment, temperature distribution across the thickness is computed by an accurate numerical method. Here, for the geometrically nonlinear analysis because of assuming large displacement, FEM is utilized according to Carrera Unified Formulation (CUF), a precise formulation and higher-order deformation theory. The principle of virtual work is employed to achieve the nonlinear equilibrium equations. The governing equations are solved by operating the arc-length method. The influence of parameters such as geometric aspect ratio, length-to-thickness ratio, boundary conditions, type of temperature distribution, and volume fraction index on the FG plates’ temperature-deflection path has been investigated. In addition, a comparison among the present study method with another reference about thermal post-buckling and also linear buckling analysis (LBA) is performed. Nonlinear post-buckling analysis shows that the bifurcation point occurs only at fully clamped boundary conditions and low aspect ratios. It proved the results of LBA in the mentioned state are verifiable. The outcomes confirm the usefulness of utilizing CUF on thermal post-buckling since the temperature curves in the figures are at a lower level compared to the similar reference.

Re-examination of thermo-mechanical buckling and postbuckling responses of sandwich plates with porous FG-GPLRC core

The current study re-examines compressive postbuckling and thermal postbuckling responses of porous sandwich plates resting on an elastic foundation. The core of the sandwich plates is made of porous graphene platelets reinforced composite (GPLRC). Compressive postbuckling of the porous sandwich plates under uniaxial compression at elevated temperature and thermal postbuckling of the same plates under uniform thermal loading are studied. The GPLRC core is made of multilayers, and each layer may have different values of porosity coefficient to obtain a piecewise functionally graded (FG) pattern. The Young’s moduli along with the shear modulus of the porous GPLRC core are estimated through a generic Halpin–Tsai model in which the porous nature of the material is taken into consideration. The material properties of the metal face sheets and the porous GPLRC core are assumed to be temperature dependent. The governing equations for the postbuckling of the porous sandwich plates reinforced by GPLs are established based on the Reddy’s third order shear deformation theory (TSDT). In the modeling, we also consider the von Kármán nonlinear strain–displacement relationships, the plate-foundation interaction and the thermal effect. A two-step perturbation approach is utilized in solving the postbuckling problem. Numerical studies are performed to compare the results obtained from the present model and the equivalent isotropic model (EIM). The results confirm that the EIM is not suitable for analyzing buckling and postbuckling of porous plates.

Static and dynamic thermal post-buckling analysis of imperfect sigmoid FG cylindrical shells resting on a non-uniform elastic foundation

In this study, an approach combining semi-analytical and analytical methods is utilized to investigate the dynamic and static thermal post-buckling behavior of imperfect sigmoid functionally graded (SFG) cylindrical shells in a thermal environment. Two types of SFG cylindrical shells consisting of ceramic-metal-ceramic (CMC) layers and metal-ceramic-metal (MCM) layers are considered. The structure is considered to rest on a non-uniform elastic foundation, and the parameters of elastic foundation used in the research are defined by using Selvadurai's methodology. Material distributions of the shells are graded through the thickness direction following a sigmoid distribution in terms of volume fraction. The nonlinear governing equation of the cylindrical shells is derived by applying the von-Kármán equation and the Donnell shell theory. Based on Galerkin's method, a discretized nonlinear governing equation is obtained for analyzing the behavior of the shells. To systematically study the dynamic thermal post-buckling response of the shells, the fourth-order P-T method is utilized. The research results obtained are verified. Moreover, to further validate the dynamic thermal post-buckling, comparisons are made with the fourth-order Runge-Kutta method. It is noted that the fourth-order P-T method has illustrated advantages over that of the fourth-order Runge-Kutta method in terms of accuracy and reliability. Furthermore, the fourth-order P-T method consumes less CPU time than that of the fourth-order Runge-Kutta method.

On the thermal buckling and postbuckling responses of temperature-dependent graphene platelets reinforced porous nanocomposite beams

In this work, thermal buckling and postbuckling behaviors of graphene platelet (GPL) reinforced porous nanocomposite beams are studied with inclusion of temperature-dependent material properties for the first time. The Halpin-Tsai micromechanics model and extended rule of mixture as well as open-cell metal foam model are applied to determine the effective properties of nanocomposites. A high order shear deformation theory with the aid of von Kármán nonlinearity is implemented to yield governing equations corresponding to thermal buckling and postbuckling problems. The numerically stable admissible functions constructed through Gram-Schmidt procedure are developed to describe end restraints of beams. An eigenvalue equation without geometric nonlinearity is solved to obtain critical buckling temperatures, while an iterative methodology is implemented to find the temperature-dependent thermal postbuckling paths of beams. An extensive numerical study is performed to check the influences of internal pores and GPL reinforcements on the thermal buckling and postbuckling responses of GPL-reinforced metal foam beams. It is found that the temperature-dependency of material properties plays important roles on thermal buckling/postbuckling behaviors of GPL-reinforced porous beams, and dispersing more GPLs on the upper and lower surfaces and fabricating more internal pores near central portion of beams can improve thermal instability-resistance capability of composite structures.

Nonlinear static simulation for thermal post-buckling analysis of composite annular system coupled with shape memory alloy fibers

The provided research deals with a mathematical formulation to achieve nonlinear statics for the sandwich annular structure’s thermal post-buckling behavior. The structure’s core is filled by graphene nanoplatelets (GPLs), while its face sheets have been made of SMA/graphite/epoxy. The first-order shear deformation (FOSD) model has been employed for modeling face sheets, and quasi-3D hybrid and quadratic kind of higher-order shear deformation (Q-3D-HSD) model is supposed for core’s in-plane transverse and displacements, respectively. The boundary conditions (BCs) and nonlinear governing equations have been modeled by employing the Hamiltonian, and they have been solved applying the generalized differential quadrature (GDQ) approach. The direct-iterative method has been provided for solving the set of equations, including highly nonlinear factors. Ultimately, the outcomes reveal that the radius ratio of outer to the inner, SMA fiber, GPLs’ configurational factors, and large amplitude would play a vital role in the annular sandwich structure’s thermal post-buckling response. Moreover, the system with GPL-O distribution pattern has maximum thermal post-buckling load while GPL-X type of distribution would result in the minimum thermal post-buckling to thermal buckling ratio for the annular structure.

A coupled nonlinear plate finite element for thermal buckling and postbuckling of piezoelectric composite plates including thermo-electro-mechanical effects

This paper investigates the thermo-electro-mechanical effects in the nonlinear response of piezoelectric laminated plates exposed to thermal and/or mechanical fields. The mechanics incorporate a) the direct piezoelectric effect through the coupling between electric-mechanical fields, b) the pyroelectric effect through the coupling between electric-thermal fields, and c) the geometric nonlinear effects due to large rotations and initial stresses. Building upon the nonlinear multifield mechanics an eight-node plate finite element is developed to discretize the generalized coupled nonlinear equations, which are subsequently linearized and iteratively solved, using the Newton-Raphson technique. Numerical evaluations demonstrate the capability of the present model to capture both of direct piezoelectric and pyroelectric effects in the nonlinear electromechanical response of piezoelectric composite plates, exposed to thermal and/or mechanical fields. The influence of direct piezoelectric and pyroelectric effects on thermal buckling and postbuckling response of various piezocomposite plates, is investigated. The possibility of pyroelectric effect to promote thermal buckling of piezocomposite plates with attached sensor patches is also quantified. The influence of thermo-electro-mechanical effects on the nonlinear response of smart composite plates, subject to either uniform or gradient temperature variations is finally illustrated.

Upheaval thermal buckling of functionally graded subsea pipelines

This study deals with the upheaval thermal buckling of functionally graded subsea pipeline subjected to thermal loadings. The temperature-dependent material properties of the functionally graded subsea pipeline are considered, which distribute through the cross-section of the pipe. The governing equations of the functionally graded subsea pipeline are derived by using the nonlinear von-Kármán assumption and Euler-Bernoulli beam theory. An accurate analytical solution is derived to evaluate the thermal post-buckling response. The proposed mathematical model is verified by comparing it with the results in the literature. Numerical results are provided to explore the influences of the temperature-dependent material properties and the volume fraction exponent. The results show that the temperature-dependent material property of the thermal expansion coefficient has a great influence on the upheaval thermal buckling behavior. In addition, both the minimum critical temperature and the maximum stress will reduce when the functionally graded subsea pipeline is employed. Thus, the functionally graded materials have potential application to the subsea pipeline transporting high-temperature oil and gas.

Isogeometric analysis for buckling and postbuckling of graphene platelet reinforced composite plates in thermal environments

This work aims at examining the buckling and postbuckling behaviour of laminated composite plate in which GPLs (graphene platelets) are functionally dispersed under various kinds of thermal loadings. For this purpose, IGA (isogeometric analysis) method of which merits encompass the precise geometric representation, smoothness of higher-order, cost-efficiency and high accuracy is taken advantage of. Shear deformable plate theory into which a hyperbolic shear shape function and the von Kármán type nonlinear kinematics are incorporated is used to derive the nonlinear equations of equilibrium for the GPL reinforced composite plate subjected to thermal gradients. On the basis of the total Lagrangian formulation, the governing equation for the thermal buckling is established and the nonlinear thermal postbuckling path is traced together with the modified Newton-Raphson iteration. Three schemes of the symmetric GPL distributions are considered. The IGA approach proposed here, by checking the performance capacity via the multiple numerical tests, is substantiated to successfully follow the nonlinear buckling and postbuckling path of the nanocomposite plate in thermal environments. The further parametric investigations highlight the impacts of the thermal load type, GPL weight fraction, GPL arrangement pattern, plate geometric parameter, plate constraint condition and surface temperature contour on the thermal buckling and postbuckling responses of the GPL strengthened composite plate.

Thermal postbuckling analysis of variable angle tow composite cylindrical panels

The advanced Automated Fiber Placement (AFP) technologies allow the fiber trajectories to be curvilinearly steered over the panel platform, which brings out a novel type of cylindrical panel structure, termed as Variable Angle Tow (VAT) composite cylindrical panel. So far, the thermal postbuckling problem of VAT composite cylindrical panels under temperature loadings remain unexplored. The aim of this article is to cover this lack by developing a methodology based on a generalized Rayleigh–Ritz formulation in the framework of the principle of thermoelastic complementary energy. The proposed panel model is based on Donnell’s shallow shell theory and accounts for von Kármán’s geometrical nonlinearity. With the aid of the Lagrange multiplier method and the introduction of Airy’s stress function, a mixed variational formula that governs the nonlinear thermoelastic problem of VAT composite cylindrical panels is derived from the principle of thermoelastic potential energy. The Rayleigh-Ritz formulation combined with the mixed variational formula is then applied to derive the simultaneous nonlinear thermoelastic equations, and the Crisfield’s arc-length method, which can cope well with snap-through or snap-back problems, is employed to trace the equilibrium path that characterizes the thermal postbuckling features. Two different modeling strategies, namely, semi-inverse method and Lagrangian multiplier method, are adopted to formulate the nonlinear thermoelastic problem of VAT composite cylindrical panels undergoing temperature change. The application of Lagrangian multiplier method overcomes some limitations inherent in semi-inverse method in terms of constructing Rayleigh-Ritz formulation, and thus provides generality to model general in-plane boundary constraint. The accuracy and robustness of the proposed Rayleigh-Ritz model are validated against finite element solutions and previously published results. Effects of geometric imperfection, curvature radius, aspect ratio and fiber orientation angle on thermal postbuckling responses of VAT cylindrical plates are discussed by several examples. The study presented here may be helpful for the preliminary design of VAT curved panels in thermal environments.

On instabilities and thermal post-buckling of the electrically annular system coupled with shape-memory alloy fibers

In this article, a mathematical derivation is made to develop a nonlinear static model for the thermal post-buckling characteristics of an annular system reinforced with graphene nanoplatelets (GPLs) and coupled with the piezoelectric actuator (PA) and shape memory alloy (SMA) under lateral loading. The matrix material is reinforced with GPLs at the nanoscale. The displacement-strain of postbuckling analysis of the functionally graded-graphene nanoplatelets reinforced composite (FG-GPLRC) annular system via higher-order shear deformation theory (HSDT) and using Von Karman nonlinearity is obtained. The governing equations and boundary conditions are formulated via the minimum total potential energy principle, and they are solved with the generalized differential quadrature method (GDQM). The direct iterative approach is presented for solving the set of equations that include highly nonlinear parameters. Finally, the results show that SMA fiber, radius ratio of outer to the inner, geometrical parameter of GPLs, applied voltage, piezoelectric thickness, and large amplitude play an essential impact on the thermal postbuckling response of the annular sandwich system. Another important consequence is that the highest thermal postbuckling to thermal buckling ratio is for the GPL-O pattern for all ranges of the large amplitude while considering the GPL-X pattern leads to the lowest thermal postbuckling to thermal buckling ratio for the smart annular system. This study’s more general conclusion is that designing the FG-GPLRC annular system should be more attention to the nonlinearity parameter.

A comprehensive computational approach for nonlinear thermal instability of the electrically FG-GPLRC disk based on GDQ method

This is a fundamental study on the buckling temperature and post-buckling analysis of functionally graded graphene nanoplatelet-reinforced composite (FG-GPLRC) disk covered with a piezoelectric actuator and surrounded by the nonlinear elastic foundation. The matrix material is reinforced with graphene nanoplatelets (GPLs) at the nanoscale. The displacement–strain of thermal post-buckling of the FG-GPLRC disk via third-order shear deformation theory and using Von Karman nonlinear plate theory is obtained. The equations of the model are derived from Hamilton’s principle and solved by the generalized differential quadrature method. The direct iterative approach is presented for solving the set of equations that includes highly nonlinear parameters. Finally, the results show that the radius ratio of outer to the inner (Ro/Ri), the geometrical parameter of GPLs, nonlinear elastic foundation, externally applied voltage, and piezoelectric thickness play an essential impact on the thermal post-buckling response of the piezoelectrically FG-GPLRC disk surrounded by the nonlinear elastic foundation. Another important consequence is that, when the effect of the elastic foundation is considered, there is a sinusoidal effect from the Ro/Ri parameter on the thermal post-buckling of the disk and this matter is true for both boundary conditions.

Closed-Form Solutions for Initial Thermal Postbuckling Response of Axially Restrained Columns

AbstractClosed-form equations for the initial thermal postbuckling response of axially restrained elastic columns with six well-known sets of rotational and lateral end restraints are presented. Th...

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

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.

Solving Post-Buckling Characteristic of Thermal-Resistance Films Attached to Glass Façade via an Optimization Method

A novel thermal-resistance film is proposed, and its thermal post-buckling behavior is studied by using the meshless method. The Von Kármán plate model together with the nonlocal theory is adopted for the formulation of the problem. The post-buckling curve of the graphene sheet (GS) film is obtained. The effect of the parameters, such as nonlocal parameters, geometric size, geometric shape and elastic medium, on the thermal post-buckling response of the GSs film is examined. It is found that it is important to design proper geometric size and geometric shape and control the radius of tensionless area. This study provides theoretical guidance for fire resistance design of thermal-resistance films attached to glass façade.

Thermal buckling and post-buckling analysis of geometrically imperfect FGM clamped tubes on nonlinear elastic foundation

• Closed form expressions are used to estimate the deflection of curved FGM tubes under thermal load . • Different types of thermal loading are considered and temperature dependency is considered. • The solution method is based on the two step perturbation technique . • Interaction of the tube with hardening/softening nonlinear elastic foundation and initial imperfection are taken into account. • The developed formulation is based on a higher order tube theory which satisfies the traction free boundary conditions. Present research deals with the thermal buckling and post-buckling analysis of the geometrically imperfect functionally graded tubes on nonlinear elastic foundation. Imperfect FGM tube with immovable clamped–clamped end conditions is subjected to thermal environments. Tube under different types of thermal loads, such as heat conduction, linear temperature change, and uniform temperature rise is analyzed. Material properties of the FGM tube are assumed to be temperature dependent and are distributed through the radial direction. Displacement field satisfies the tangential traction free boundary conditions on the inner and outer surfaces of the FGM tube. The nonlinear governing equations of the FGM tube are obtained by means of the virtual displacement principle. The equilibrium equations are based on the nonlinear von Kármán assumption and higher order shear deformation circular tube theory. These coupled differential equations are solved using the two-step perturbation method. Approximate solutions are provided to estimate the thermal post-buckling response of the perfect/imperfect FGM tube as explicit functions of the various thermal loads. Numerical results are provided to explore the effects of different geometrical parameters of the FGM tube subjected to different types of thermal loads. The effects of power law index, springs stiffness of elastic foundation, and geometrical imperfection parameter of tube are also included.

Thermal post-buckling analysis of functionally graded material structures using a modified FSDT

The thermal post-buckling responses of Functionally Graded Material shell structures are reported in this paper. Geometrically nonlinear analysis based on a modified First order Shear Deformation Theory are proposed. The modified theory takes into account the shear strains with a parabolic shape function and it verifies a zero shear stresses condition at the top and bottom surfaces. For the numerical computation, four nodes shell elements are implemented. The large displacement is described by Green–Lagrange nonlinear strains. Moreover, it is assumed that the shell structures are exposed to uniform, linear and nonlinear temperature distributions through the thickness direction. The thermal and the mechanical properties are described according to a power law distribution and either temperature-independent or temperature-dependent material properties are considered. Two numerical examples of functionally graded plates and cylindrical shells are presented to highlight the effectiveness and the accuracy of the present finite element procedure. The effect the geometrical parameters, the volume fraction index and boundary conditions on nonlinear responses are performed.

NURBS-based isogeometric thermal postbuckling analysis of temperature dependent graphene reinforced composite laminated plates

A thermal postbuckling analysis for composite laminated plates reinforced with graphene sheets is performed in this research. All of the thermomechanical properties of the composite media are assumed to be temperature dependent. Volume fraction of the graphene in each layer is assumed to be different which results in a piecewise functionally graded plate. Based on the third order shear deformation plate theory of Reddy, the total strain energy of the plate is obtained. Composite laminated plate is assumed to be under uniform temperature rise. Properties of the graphene reinforced composite media are estimated by means of a refined Haptin-Tsai approach which contains efficiency parameters to capture the size dependency of the constituents. Afterwards, a non-uniform rational B-spline (NURBS) based isogeometric finite element method is implemented to study the thermal postbuckling response of the graphene reinforced composite laminated plates. Thermally induced postbuckling curves of the composite plate reinforced by graphene are provided for different functionally graded patterns, aspect ratios, side to thickness ratios and boundary conditions. It is shown that, FG-X pattern of graphene reinforcement results in the highest critical buckling temperature and the lowest postbuckling deflection.

Semi-analytical approach for thermal buckling and postbuckling response of rectangular composite plates subjected to localized thermal heating

A semi-analytical solution of thermal buckling and postbuckling equilibrium paths of rectangular composite plates subjected to localized thermal heating over the plate area with uniform temperature rise through its thickness is reported here. The plate is either simply supported or clamped for out-of-plane displacements and immovable for in-plane displacements. The thermal buckling of the composite plate subjected to localized thermal loading is solved in two steps as the prebuckling stress distributions within the plate are not known a priori. In the first step the explicit analytical expressions for these in-plane prebuckling stress distributions within the composite plate are developed by solving the thermoelasticity problem using Airy’s stress function. Using the computed in-plane stress distributions within the plate, the plate nonlinear stability equations are formulated using a variational principle. First-order shear deformation theory incorporating von-Karman geometric nonlinearity is used to model the plate. The generalized differential quadrature method in conjunction with the modified Newton–Raphson method is used to solve the nonlinear governing partial differential equations for nonlinear thermal stability of composite plates. The influence of aspect ratio, area of heated region, and the boundary conditions on the critical buckling temperature and equilibrium paths are examined.

Nonlinear stability analysis of a perforated FGM plate under thermal load

ABSTRACTThe nonlinear thermal stability analysis of a perforated Ni/Al2O3 FGM plate with temperature-dependent thermoelastic material properties is carried out through a finite element approach based on the first-order shear deformation theory and the von-Karman's nonlinear kinematics. The thermoelastic material properties of the FGM plate are varied in the thickness direction, and the Mori-Tanaka homogenization scheme is applied to evaluate these properties. After validating the results of the current formulation, parametric studies are conducted to examine the effects of temperature-dependent material properties, material in-homogeneity, cutout shape and size, edge boundary conditions, aspect and slenderness ratios on the thermal buckling and postbuckling response of perforated FGM plate.

Thermal postbuckling analysis of anisotropic laminated beams with tubular cross-section based on higher-order theory

Buckling of composite riser assumed beam structure is one of numerous engineering challenges in deep water pipeline design. Thermal postbuckling analysis of shear deformable anisotropic laminated composite beams with tubular cross-section subjected to uniform, linear and non-linear temperature distribution through the thickness resting on a two-parameter elastic foundation is presented. The material of each layer for the composite beam with tubular section is assumed to be linearly elastic and fiber-reinforced. The governing equations are introduced by using high-order shear deformation beam model with a von Kármán-type of kinematic nonlinearity. Composite beams with clamped–clamped, clamped–hinged, and hinged–hinged boundary conditions are considered. A numerical solution for nonlinear partial-integral differential form in terms of the transverse deflection by using Galerkin׳s method is employed to determine the buckling temperatures and postbuckling equilibrium paths of anisotropic laminated beams with different types of temperature distribution through the thickness. The numerical illustration concern the thermal postbuckling response of laminated beams with different types of boundary conditions, ply arrangements (lay-ups), geometric and physical properties. The results reveal that the geometric and physical properties, temperature dependent properties, initial geometry imperfection, boundary conditions and elastic foundation have a significant effect on thermal postbuckling behavior of anisotropic laminated composite tubular beams.