Thermal Postbuckling Research Articles

Thermal buckling and postbuckling of edge-cracked functionally graded multilayer graphene nanocomposite beams on an elastic foundation

This paper investigates thermal buckling and postbuckling behaviors of functionally graded graphene nanoplatelet (GPL)-reinforced composite multilayer beams containing an open edge crack and resting on a Pasternak-type elastic foundation based on the first-order shear deformation beam theory including von Kármán geometric nonlinearity. The material properties of functionally graded GPL-reinforced composites (GPLRCs), which exhibit piece-wise variation along the thickness direction, are evaluated using micromechanics based models. The bending stiffness of the cracked section is estimated by the rotational spring model. The obtained nonlinear partial differential equations of equilibrium are discretized by the differential quadrature method, and then an iterative method is used to obtain the thermal buckling loads and postbuckling load-deflection curves. Detailed parametric studies are conducted to investigate the effects of crack length, GPL distribution pattern, GPL weight fraction, GPL length-to-width and length-to-thickness ratios, boundary conditions, and foundation stiffnesses on the thermal buckling loads and postbuckling response of the cracked GPLRC beams.

Thermal buckling and postbuckling of CNT-reinforced composite cylindrical shell surrounded by an elastic medium with tangentially restrained edges

Buckling and postbuckling behavior of thin composite cylindrical shells reinforced by carbon nanotubes (CNTs), surrounded by elastic media and exposed to uniform temperature rise, are investigated in this article. CNTs are reinforced into isotropic matrix phase through uniform distribution or functionally graded distributions across the thickness direction. Material properties are assumed to be temperature dependent, and effective elastic moduli of CNT-reinforced composite are determined according to extended rule of mixture. Formulations are based on the classical thin shell theory taking Von Karman–Donnell nonlinearity, surrounding elastic media and elastic constraints of boundary edges into consideration. Multi-term solutions of deflection and stress function are assumed to satisfy simply supported boundary condition, and Galerkin method is applied to obtain nonlinear relation of thermal load and deflection. An iteration algorithm is used to determine buckling temperatures and postbuckling paths. Numerical examples are given to analyze the effects of volume fraction and distribution type of CNTs, geometrical parameters, degree of tangential edge constraint, buckling mode, and surrounding elastic media on the buckling temperatures and postbuckling strength of thermally loaded nanocomposite cylindrical shells.

Nonlinear Vibration of Thermally Postbuckled FG-GRC Laminated Beams Resting on Elastic Foundations

Investigated herein are the small- and large-amplitude vibrations of a thermally postbuckled graphene-reinforced composite (GRC) laminated beam supported by an elastic foundation. The piecewise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the beam. The temperature-dependent material properties of functionally graded graphene-reinforced composites (FG-GRCs) are estimated through the extended Halpin–Tsai micromechanical model. The nonlinear governing differential equations are derived from the higher-order shear deformation beam theory and the von Kármán-type strain–displacement relationships. The thermal effect, the beam–foundation interaction and the initial deflection caused by thermal postbuckling are also included. A two-step perturbation approach is applied to determine the thermal postbuckling equilibrium paths as well as the nonlinear vibration solutions for the FG-GRC laminated beams. Results are presented to demonstrate the nonlinear vibration responses of thermally postbuckled FG-GRC laminated beams under a uniform temperature field. The effects of the FG reinforcement patterns and the foundation stiffness on the nonlinear vibration responses of FG-GRC laminated beams are examined and discussed.

Vibration of thermally postbuckled FG-GRC laminated plates resting on elastic foundations

This paper investigates the small- and large-amplitude vibrations of thermally postbuckled graphene-reinforced composite (GRC) laminated plates resting on elastic foundations. The piecewise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the plate. The anisotropic and temperature-dependent material properties of the FG-GRC layers are estimated through the extended Halpin–Tsai micromechanical model. Based on the Reddy's higher order shear deformation plate theory and the von Kármán strain–displacement relationships, the motion equations of the plates are derived. The foundation support, the thermal effect, and the initial deflection caused by thermal postbuckling are also included in the derivation. A two-step perturbation approach is applied to determine the thermal postbuckling equilibrium paths as well as the nonlinear vibration solutions for the FG-GRC laminated plates. The numerical illustrations concern small- and large-amplitude vibration characteristics of thermally postbuckled FG-GRC laminated plates under a uniform temperature field. The effects of graphene reinforcement distributions and foundation stiffnesses on the vibration responses of FG-GRC laminated plates are examined in detail.

Analytical layerwise solution of nonlinear thermal instability of SMA hybrid composite beam under nonuniform temperature condition

Thermal buckling and post-buckling behavior of composite beam reinforced with shape memory alloy (SMA) wires under nonuniform temperature distribution is explored. Thermo-mechanical behavior of SMA wires is formulated by using the one-dimensional Brinson SMA model. Considering von Karman strain–displacement relation, corresponding nonlinear governing equations are obtained and solved analytically. Heat conduction equation is employed and through-the-thickness temperature distribution is obtained by discretization scheme of layerwise method. Influence of SMA-wire positioning across the thickness, temperature distribution, SMA wire pre-straining level and volume fraction of SMAs upon the thermal buckling and post buckling of reinforced beam are examined and discussed in detail.

Thermal post-buckling of sandwich beams with functionally graded negative Poisson's ratio honeycomb core

This paper investigates the thermal post-buckling behavior of sandwich beams with functionally graded (FG) negative Poisson's ratio (NPR) honeycomb cores. Two symmetric FG configurations of re-entrant honeycomb cores along the beam thickness direction are proposed for the first time. The material properties of both face sheets and core of the sandwich beams are assumed to be temperature-dependent. The thermal post-buckling behavior and the variation of effective Poisson's ratio (EPR) of the sandwich beam in the large deflection region are studied by using 3D full scale finite element simulations. Numerical results are presented for the sandwich beams with FG-NPR honeycomb core under a uniform temperature field, from which results for the same sandwich beam with uniform distributed NPR honeycomb core are obtained as a comparator. The EPR-deflection curves are obtained for the first time, and the results reveal that greater bending stiffness could bring about higher EPR-deflection curve. The effects of functionally graded configurations, boundary conditions, facesheet-to-core thickness ratios, cell wall-to-facesheet thickness ratios and length-to-thickness ratios on the thermal post-buckling load-deflection curves and EPR-deflection curves of sandwich beams are discussed in detail.

Isogeometric thermal postbuckling of FG-GPLRC laminated plates

An analysis on thermal buckling and postbuckling of composite laminated plates reinforced with a low amount of graphene platelets is performed in the current investigation. It is assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the composite media. Elastic properties of the nanocomposite media are obtained by means of the modified Halpin-Tsai approach which takes into account the size effects of the graphene reinforcements. By means of the von Karman type of geometrical nonlinearity, third order shear deformation theory and nonuniform rational B-spline (NURBS) based isogeometric finite element method, the governing equations for the thermal postbuckling of nanocomposite plates in rectangular shape are established. These equations are solved by means of a direct displacement control strategy. Numerical examples are given to study the effects of boundary conditions, weight fraction of graphene platelets and distribution pattern of graphene platelets. It is shown that, with introduction of a small amount of graphene platelets into the matrix of the composite media, the critical buckling temperature of the plate may be enhanced and thermal postbuckling deflection may be alleviated.

Thermal postbuckling of shear deformable CNT-reinforced composite plates with tangentially restrained edges and temperature-dependent properties

Buckling and postbuckling behaviors of moderately thick composite plates reinforced by single-walled carbon nanotubes (SWCNTs), rested on elastic foundations and subjected to two types of thermal loading are investigated in this article. Carbon nanotubes (CNTs) are reinforced into isotropic polymer matrix according to functional rules in which volume fractions of constituents are graded in the thickness direction. Material properties of constituents are assumed to be temperature-dependent and effective properties of nanocomposite are estimated by extended rule of mixture. Formulations are based on first-order shear deformation theory taking von Karman nonlinearity, initial geometrical imperfection, tangential constraints of edges, and two-parameter elastic foundation into consideration. Approximate solutions are assumed to satisfy simply supported boundary conditions and Galerkin method is applied to derive nonlinear temperature–deflection relations from which buckling temperatures and postbuckling equilibrium paths are determined by an iteration algorithm. Novel findings of the present study are that deteriorative influences of temperature-dependent properties on the postbuckling behavior become more serious as plate edges are partially movable, CNT volume fraction is higher, elastic foundations are stiffer, plates are thicker, and/or temperature linearly changed across the thickness.

Thermal Buckling and Postbuckling Analysis of Functionally Graded Concrete Slabs with Initial Imperfections

This paper proposes a novel functionally graded (FG) concrete slab and investigates its thermal buckling and postbuckling performance using the finite-element (FE) method. The concrete slab consists of three homogeneous thick layers — a fiber-reinforced concrete layer, a geopolymer concrete layer, and a plain Portland cement (PPC) layer — with a thin FG layer between the thick layers. The mechanical properties of the thin FG layers are exponentially graded across the thickness direction. The effects of initial imperfection, the self-weight of the slab, and the friction between the slab and rigid foundation are considered in the analysis. The FE model is validated against the results reported in the literature. A comprehensive parametric study is conducted to examine the effects of the thickness and volume fraction index of the FG layer, initial imperfection, self-weight, friction, and slab slenderness ratio on the thermal buckling and postbuckling behaviors of the concrete slab. The numerical results demonstrate that the proposed FG slab exhibits remarkably better buckling and postbuckling resistance than a conventional PPC concrete slab and that the influences of both self-weight and friction are important and cannot be neglected.

Dynamic thermal buckling and postbuckling of clamped–clamped imperfect functionally graded annular plates

Dynamic thermal buckling and postbuckling of imperfect functionally graded material (FGM) annular plates are investigated based on the nonlinear plate theory. The transient temperature fields of the FGM annular plates under dynamic thermal loadings are obtained by Laplace transform in combination of power series method according to the theory of Fourier heat conduction. The nonlinear dynamic equations of large axisymmetric deformations are numerically solved by series expansions and Runge–Kutta method. The critical bucking and dynamic postbuckling responses are predicted by the maximum deflections of the FGM annular plates with positive or negative initial geometric imperfections. The effects of the loads, the material gradient and the initial geometric imperfections on the dynamic responses and the buckling critical temperatures of the FGM annular plates are analyzed in detail.

A comparative study on the postbuckling behaviour of circular plates

Materials/metals, polymers, reinforced plastics, composites, functionally graded materials, ceramics and nano materials find significant place in many engineering and scientific applications. The material structural members such as plates and beams are subjected to structural deformation owing to several external and internal factors such as temperature, hysteresis and so on. Therefore, to analyze the structural deformation of these material structures, its nonlinear behavior modeling is relevant. The rapid progress in high speed computing devices resolve research problems involving material non-linearity behavior such as post-buckling behavior in varied structures. The post-buckling behavior of commonly used structural elements especially plates under thermal and mechanical loads are of great practically important research topic. Therefore, a comparative study on the thermal post buckling behavior of plates by various numerical methods such as finite element formulation, shear deformation theory, Rayleigh – Ritz method, iteration methods and so on will be helpful to obtain a cutting edge on numerical solutions and their applications in varied engineering applications. This paper reports a novel mathematical formulation involving substitution method to evaluate and compare the thermal post buckling load carrying capacity of circular plates with minimal errors.

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

ABSTRACTPresent 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.

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.

Nonlinear thermal torsional post-buckling of carbon nanotube-reinforced composite cylindrical shell with piezoelectric actuator layers surrounded by elastic medium

The paper focuses on thermal torsional post buckling of functionally graded carbon nanotube reinforced composite cylindrical shells with sur-bonding piezoelectric layers and embedded in an elastic medium. The distribution of reinforcements through the thickness of the shells is considered to be uniform and functionally graded. The basic equations using geometrically nonlinearity in von Karman-Donnell sense within the classical thin shell theory are established. The torsional post-buckling behavior of the piezoelectric functionally graded carbon nanotubes reinforced composite (FG-CNTRC) shells is analyzed with using the Airy's stress function and the Galerkin's method, in which a three-term approximate solution of the deflection of the shell is assumed. Effects of thermal environment, CNT volume fraction, piezoelectric layers (the thickness and the constant voltage), distribution type of the reinforcement, dimensional parameters and Winkler and Pasternak foundation are investigated in this paper. Numerical and graphical results show that the carbon nanotube volume fraction and the piezoelectric layers as well as the elastic foundation and the thermal loads have significantly influenced on the torsional postbuckling behavior of nanocomposite shells.

Postbuckling and nonlinear vibration of composite Laminated trapezoidal plates

The thermal effects on the buckling, postbuckling and nonlinear vibration behaviors of composite laminated trapezoidal plates are studied. Aiming at the complex plate structure and to simulate the temperature distribution of the plate, a finite element method (FEM) is applied in this paper. In the temperature model, based on the thermal diffusion equation, the Galerkin\'s method is employed to establish the temperature equation of the composite laminated trapezoidal plate. The geometrical nonlinearity of the plate is considered by using the von Karman large deformation theory, and combining the thermal model and aeroelastic model, Hamilton\'s principle is employed to establish the thermoelastic equation of motion of the composite laminated trapezoidal plate. The thermal buckling and postbuckling of the composite laminated rectangular plate are analyzed to verify the validity and correctness of the present methodology by comparing with the results reported in the literature. Moreover, the effects of the temperature with the ply-angle on the thermal buckling and postbuckling of the composite laminated trapezoidal plates are studied, the thermal effects on the nonlinear vibration behaviors of the composite laminated trapezoidal plates are discussed, and the frequency-response curves are also presented for the different temperatures and ply angles.

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

ABSTRACTIn 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 of FGM circular plates with in-plane elastic restraints

Based on von Karman’s plate theory, the axisymmetric thermal buckling and post-buckling of the functionally graded material (FGM) circular plates with inplane elastic restraints under transversely non-uniform temperature rise are studied. The properties of the FGM media are varied through the thickness based on a simple power law. The governing equations are numerically solved by a shooting method. The results of the critical buckling temperature, post-buckling equilibrium paths, and configurations for the in-plane elastically restrained plates are presented. The effects of the in-plane elastic restraints, material property gradient, and temperature variation on the responses of thermal buckling and post-buckling are examined in detail.

Thermal buckling and postbuckling of functionally graded graphene nanocomposite plates

This paper deals with the thermal buckling and postbuckling of functionally graded multilayer nanocomposite plates reinforced with a low content of graphene platelets (GPLs). It is assumed that GPL reinforcements are randomly oriented and uniformly dispersed in each individual GPL-reinforced composite (GPLRC) layer but the concentration follows a layer-wise variation across the plate thickness. The modified Halpin-Tsai micromechanics model that takes into account the GPL geometry effect is adopted to estimate the effective Young's modulus of GPLRC layers. Within the framework of the first-order shear deformation theory, the nonlinear governing equations are derived by applying the principle of virtual displacements and then solved by using a differential quadrature-based iteration technique. Parametric studies are conducted to examine the influences of GPL distribution pattern, concentration and geometry, as well as in-plane force on the thermal buckling and postbuckling behaviours. Our results show that distributing more GPLs near the surface layers is capable of reinforcing the thermal buckling and postbuckling performances of GPLRC plates. Whether the thermal buckling and postbuckling resistance increases or decreases with the increases in GPL weight fraction, aspect ratio and width-to-thickness ratio is highly dependent on the GPL distribution pattern.

Thermal buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates resting on elastic foundations

This paper presents the modeling and analysis for the thermal postbuckling of graphene-reinforced composite laminated plates resting on an elastic foundation and subjected to in-plane temperature variation. A micromechanical model is used to estimate the temperature-dependent material properties of the graphene-reinforced composites (GRCs). Piece-wise functionally graded (FG) GRC layers along the thickness direction of a plate is considered in this study. Employing the higher order shear deformation plate theory, the governing equations for FG-GRC plates are derived and the effects of plate-foundation interaction and temperature variation are included in the modeling. A two-step perturbation technique is applied to obtain the buckling temperature and the thermal postbuckling load-deflection curves for perfect and imperfect FG-GRC laminated plates. The results show that the buckling temperature as well as thermal postbuckling strength of the plates can be increased as a result of the functionally graded graphene reinforcement for the plates.

Exact solution of thermal buckling and post buckling of composite and SMA hybrid composite beam by layerwise theory

The thermal buckling and post-buckling behavior of laminated composite beam reinforced with Shape Memory Alloy (SMA) wires is explored. The thermo-mechanical behavior of embedded SMA wires is formulated based on one-dimensional Brinson SMA model. Using a variational approach, the Layer-Wise Theory (LWT) of beams is employed to develop the governing equations. The von Karman strain–displacement relations are used to account for the large deflection effects. A closed-form solution of the resulting systems of nonlinear equations is represented. Exemplary case studies are assessed by comparing with available literature. Considering symmetric and antisymmetric layups, the influence of thickness/span ratio, degree of anisotropy, SMA pre-strain and volume fraction upon the buckling and post-buckling of thick composite and hybrid composite beams are examined and discussed in detail.