Thermal Postbuckling Equilibrium Path Research Articles

Mechanical and thermal postbuckling of functionally graded graphene origami-enabled auxetic metamaterials plates

Postbuckling under high-temperature biaxial compression and thermal postbuckling under uniform thermal loading of functionally graded (FG) graphene origami (GOri)-enabled auxetic metamaterials (GOEAMs) plates are investigated. The plates consist of multiple layers of GOEAMs with varying levels of GOri content in the layers across the direction of the thickness. The nonlinear governing equations of the plate are based on the first-order shear deformation theory (FSDT) and the von Kármán type nonlinear strain–displacement relationship. The mechanical and thermal postbuckling equilibrium paths of FG-GOEMA plates on the elastic foundation are determined using the two-step perturbation method. The effects of the elastic foundation, temperature, GOri content, GOri distribution pattern, and degree of GOri folding on the mechanical and thermal postbuckling of FG-GOEMA plates are discussed.

Influence of couple stress size dependency in thermal instability of porous functionally graded composite microplates having different central cutouts

In the present study, the isogeometric numerical solving process incorporating non-uniform rational B-splines is put to use to analyze the size-dependent thermal postbuckling behavior of porous functionally graded (FG) microplates having a central cutout with different shapes. Accordingly, the modified couple stress continuum elasticity is employed within the framework of a hybrid-type quasi-3D higher-order plate theory to take the through-thickness deformations into consideration by only four variables. On the basis of a refined power-law function together with the Touloukian scheme, the porosity-dependent as well as temperature-dependent material properties are achieved. The couple stress-based thermal postbuckling equilibrium paths are acquired corresponding to various geometrical and material parameters and different boundary conditions. It is found that the gap between thermal postbuckling equilibrium paths relevant to various patterns of the porosity dispersion is a bit higher for the couple stress-based case than the classical one. Furthermore, it is indicated that a central cutout causes to change the trend of the load-deflection response that leads to decrease the initial thermal postbuckling strength, while it enhances the microplate strength in deep thermal postbuckling region.

Isogeometric thermal postbuckling analysis of porous FGM quasi-3D nanoplates having cutouts with different shapes based upon surface stress elasticity

In the current investigation, a surface elastic-based three-dimensional (3D) nonlinear formulation is provided to explore the thermal postbuckling characteristics of porous composite nanoplates made of a functionally graded material (FGM) having a central cutout with different shapes. The surface stress factors are taken into the formulation by choosing the Gurtin-Murdoch theory of elasticity within the framework of a hybrid-type quasi-3D higher-order plate theory. With the aid of the isogeometric solving process using non-uniform rational B-splines as the basis functions, higher-order approximation with a precise geometric modelling of the structure is achieved. In the context of a refined power-law function together with the Touloukian scheme, the porosity-dependent as well as temperature-dependent material properties are achieved. It is portrayed that for a higher value of the material gradient index, the role of surface stress type of size dependency in the thermal postbuckling of porous FGM nanoplates becomes more important. Furthermore, it is deduced that by going to deeper part of the postbuckling regime which results in a higher value of the maximum deflection, the increment in the associated temperature rise caused by the surface stress size effect reduces. Also, it is found that by considering a higher value of the maximum deflection and moving to deeper region of the postbuckling domain, the existence of a central cutout leads to change the trend of the thermal postbuckling equilibrium path especially for a bigger central cutout.

Thermal Postbuckling of Temperature-Dependent Functionally Graded Nanocomposite Annular Sector Plates Reinforced by Carbon Nanotubes

In this study, the thermal buckling and postbuckling of functionally graded (FG) nanocomposite annular sector plates reinforced by carbon nanotubes (CNTs) are numerically analyzed. The effective material properties of FG nanocomposite are temperature-dependent (TD) and evaluated via the modified micromechanical method and rule of mixture. Based on the higher-order shear deformation theory (HSDT) and using the principle of virtual work and variational differential quadrature (VDQ) approach, the unified weak form of discretized nonlinear governing equilibrium equations is derived. Then, by using the linear part of equations and solving the derived eigenvalue problem, the critical temperature rise and associated mode shapes are obtained, which are used as the initial guess in solving the nonlinear thermal postbuckling problem. The pseudo-arc-length method and an iterative solver are employed to obtain the nonlinear thermal postbuckling equilibrium path of the FG nanocomposite annular sector plates. The influences of geometrical parameters, boundary conditions (BCs), CNT volume fraction, and CNT distribution pattern on the critical temperature rise and thermal postbuckling behavior of the FG nanocomposite annular sector plates are evaluated and discussed. Also, comparisons are made between the results considering the TD and temperature-independent (TID) properties. It is demonstrated that for higher values of sector angle, the effect of sector angle on the critical temperature rise and thermal postbuckling path is negligible. Moreover, by increasing the sector angle, the effect of BCs of straight edges vanishes, and the critical temperature rise and thermal postbuckling curves of for BCs of CSCS and SCSC approach those for CCCC and SSSS ones.

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.

Thermal buckling and postbuckling behavior of FG-GRC laminated cylindrical shells with temperature-dependent material properties

Thermal postbuckling analysis is presented for graphene-reinforced composite (GRC) laminated cylindrical shells under a uniform temperature field. The GRC layers are arranged in a functionally graded (FG) graphene reinforcement pattern by varying the graphene volume fraction in each GRC layer. The GRCs possess temperature dependent and anisotropic material properties and the extended Halpin–Tsai model is employed to evaluate the GRC material properties. The governing equations are based on a higher order shear deformation shell theory and include the von Karman-type kinematic nonlinearity and the thermal effects. A singular perturbation method in conjunction with a two-step perturbation approach is applied to determine the thermal postbuckling equilibrium path for a GRC shell with or without geometric imperfection. An iterative scheme is developed to obtain numerical thermal buckling temperatures and thermal postbuckling load–deflection curves for the shells. The results reveal that the FG-X piece-wise FG graphene distribution can enhance the thermal postbuckling capacity of the shells when the shells are subjected to a uniform temperature loading.

Thermal postbuckling behavior of FG-GRC laminated cylindrical panels with temperature-dependent properties

The current study deals with thermal postbuckling behavior of graphene-reinforced composite (GRC) laminated cylindrical panels resting on elastic foundations. The GRC layers of the panel are arranged in a piece-wise functionally graded (FG) distribution along the thickness direction, and each layer of the panel contains different volume fractions of graphene reinforcement. The temperature dependent material properties of GRCs are estimated by the extended Halpin–Tsai micromechanical model with graphene efficiency parameters being calibrated against the GRC material properties obtained from the molecular dynamics simulations. The nonlinear governing equations for the thermally-loaded GRC laminated cylindrical panels are derived based on the higher order shear deformation theory and include the geometric nonlinearity effects in the sense of the von Kármán nonlinear kinematic assumptions. The panel-foundation interaction and thermal effects are also considered. The thermal postbuckling equilibrium paths for the perfect and geometrically imperfect GRC laminated cylindrical panels are obtained by applying a singular perturbation method in conjunction with a two-step perturbation approach. An iterative scheme is developed to obtain the numerical thermal postbuckling solutions of the panels. We observe that the piece-wise functionally graded distribution of graphene reinforcement can enhance the thermal postbuckling strength of the GRC laminated cylindrical panel under a uniform temperature field.

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.

Enhancement of non-linear thermal stability of temperature dependent laminated beams with graphene reinforcements

Present investigation deals with the thermal postbuckling behaviour of a composite laminated beam where graphene is used as reinforcement of each lamina. The composite laminated beam may be piecewise functionally graded by changing the volume fraction of graphene in each lamina. Mechanical properties in each layer are obtained according to the modified Halpin-Tsai approach which contains efficiency parameters to capture the size dependency of the nanocomposite properties. Thermomechanical properties of the matrix and reinforcements are assumed to be temperature dependent. Beam is resting on an elastic foundation which acts in compression as well as in tension. The first order shear deformation beam theory and von Kármán type of geometrical nonlinearity are used as the basic assumptions to establish the total strain energy of the system. Beam is subjected to uniform temperature elevation. With the aid of conventional Ritz method and the simple polynomials as the basic functions, the matrix representation of the governing equations is derived. The adopted solution method may be used for arbitrary combinations of boundary conditions. The obtained system of equations is nonlinear in terms of both temperature and displacements. An iterative displacement control strategy is proposed to extract the thermal postbuckling curves of the beam resting on a conventional elastic foundation. Numerical results are given to discuss the effects of graphene distribution, stacking sequence, boundary conditions, side to thickness ratio and foundation stiffness on critical buckling temperature and thermal postbuckling equilibrium path of the beam. It is shown that, through a piecewise functionally graded distribution of graphene in matrix, critical buckling temperature may be enhanced significantly and thermal postbuckling deflection may be alleviated.

Nonlinear bending and thermal postbuckling of functionally graded graphene-reinforced composite laminated beams resting on elastic foundations

This paper presents an investigation on the nonlinear bending and thermal postbuckling behaviors of nanocomposite beams in thermal environments and supported by an elastic foundation. Graphene-reinforced composite (GRC) material is used for the beams with piece-wise functionally graded (FG) graphene reinforcement along the thickness direction of the beams. A refined micromechanical model is applied to estimate the material properties of GRCs and the effect of temperature is included in the model. The governing equations of a GRC beam are based on a higher third order beam theory with the effect of temperature variation and foundation interaction and the von Kármán geometric nonlinear strain terms are also considered. Applying a two-step perturbation technique, the governing equations of the GRC beams are solved to determine the nonlinear bending load-deflection curves and the thermal postbuckling equilibrium paths of the beams. The effects of the graphene reinforcement distribution, laminate layer stacking sequence, temperature variation and foundation stiffness on the nonlinear bending and thermal postbuckling behaviors of the beams are discussed in details.

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.

Imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite beams

This paper investigates the imperfection sensitivity of thermal post-buckling behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams subjected to in-plane temperature variation. The material properties of FG-CNTRCs are assumed to be graded in the thickness direction and temperature-dependent. A generic imperfection function is used to model various possible imperfections, including sine type, global and localized imperfections. The governing equations are derived based on the first-order shear deformation beam theory and von-Kármán geometric nonlinearity. The differential quadrature method in conjunction with modified Newton–Raphson technique is employed to determine the thermal post-buckling equilibrium path of imperfect FG-CNTRC beams. Thermal buckling is treated as a subset problem. A parametric study is conducted to examine the effects of imperfection mode, half-wave number, location and amplitude on their thermal post-buckling performance. The influences of distribution pattern and volume fraction of carbon nanotubes, boundary conditions and slenderness ratio are discussed as well. The results indicate that the thermal post-buckling is highly sensitive to the imperfection mode, half-wave number, location as well as its amplitude. It is also shown that the clamped-clamped FG-CNTRC beam is more sensitive to imperfections than those with other boundary conditions whereas other parameters do not substantially affect the imperfection sensitivity of thermal post-buckling behaviour.

Thermal post-buckling behavior of imperfect temperature-dependent sandwich FGM plates resting on Pasternak elastic foundation

In this paper, post-buckling behavior of sandwich plates with functionally graded (FG) face sheets under uniform temperature rise loading is examined based on both sinusoidal shear deformation theory and stress function. It is supposed that the sandwich plate is in contact with an elastic foundation during deformation, which acts in both compression and tension. Thermo-elastic non-homogeneous properties of FG layers change smoothly by the variation of power law within the thickness, and temperature dependency of material constituents is considered in the formulation. In the present development, Von Karman nonlinearity and initial geometrical imperfection of sandwich plate are also taken into account. By employing Galerkin method, analytical solutions of thermal buckling and postbuckling equilibrium paths for simply supported plates are determined. Numerical examples presented in the present study discuss the effects of gradient index, sandwich plate geometry, geometrical imperfection, temperature dependency, and the elastic foundation parameters.

Thermal Buckling and Postbuckling Analysis of Functionally Graded Carbon Nanotube-Reinforced Composite Beams

Thermal buckling and postbuckling of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams are investigated in this paper based on Timoshenko beam theory within the framework of von Kármán geometric nonlinearity. The material properties of FG-CNTRCs are assumed to be temperature-dependent and vary in the beam thickness direction. The governing equations are derived by employing Hamilton’s principle then discretized by using differential quadrature (DQ) method. An iterative scheme is used to obtain the critical buckling temperature and nonlinear thermal postbuckling equilibrium path of the FG-CNTRC beam. Numerical results are presented for FG-CNTRC beams hinged or clamped at both ends, with particular focuses on the effects of the volume fraction of carbon nanotubes (CNTs), slenderness ratio, and end supports on the thermal buckling and postbuckling characteristics.

Study on the thermal buckling and post-buckling of metallic sub-stiffening structure and its optimization

Thermal buckling and post-buckling behaviors of sub-stiffened panels were analyzed. The study was carried out by constructing an optimization processes of metallic sub-stiffened panel under the framework of Multidisciplinary software ModelCenter and Finite element software ABAQUS. The optimal design parameters of sub-stiffened panels for three different kinds of layout were acquired with Particle Swarm Optimization algorithm. The optimal results showed that, compared to the conventional stiffened panel with equivalent mass designs, the critical thermal buckling temperatures of prismatic sub-stiffened panel, grid sub-stiffened panel and curvilinear sub-stiffened panel increased by 63.1, 57.1 and 70.5 %, respectively. The thermal post-buckling equilibrium paths of conventional stiffened panel and sub-stiffened panels were also compared by using Riks method. The results indicated that the introduction of sub-stiffening to the conventional stiffened panel can significantly improve the panel’s thermal buckling load and thermal post-buckling behavior, which demonstrates that sub-stiffened structures have a promising application prospect in the field of structural engineering.

Nonlinear bending and thermal postbuckling of functionally graded fiber reinforced composite laminated beams with piezoelectric fiber reinforced composite actuators

This paper investigates the nonlinear bending in thermal environments and thermal postbuckling of hybrid laminated beams with piezoelectric fiber reinforced composite (PFRC) actuators resting on an elastic foundation. The beam is made of fiber-reinforced composites (FRCs) with the reinforcement being distributed either uniformly (UD) or functionally graded (FG) of piece-wise type along the thickness of the beam. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The beam–foundation interaction and thermo-piezoelectric effects are also included, and the material properties of both FRCs and PFRCs are estimated through a micromechanical model and are assumed to be temperature dependent. The nonlinear bending load-deflection curves and the thermal postbuckling equilibrium paths of hybrid laminated beams are determined by means of a two-step perturbation approach. The effects of the material property gradient, temperature variation, applied voltage, stacking sequence as well as the foundation stiffness on the nonlinear bending and thermal postbuckling behaviors of the hybrid laminated beams are investigated through a comprehensive parametric study.

Nonlinear analysis of functionally graded fiber reinforced composite laminated beams in hygrothermal environments, Part II: Numerical results

In this part, the extensive parametric studies performed are reported and numerical results are presented for the nonlinear vibration, nonlinear bending and thermal postbuckling of uniformly distributed and functionally graded fiber reinforced cross-ply and angle-ply laminated beams resting on Pasternak elastic foundations under different sets of hygrothermal environmental conditions. The numerical results reveal that a functionally graded reinforcement has a significant effect on the nonlinear vibration characteristics, nonlinear bending behaviors, and thermal postbuckling behaviors of fiber reinforced composite (FRC) laminated beams. The results show that the temperature/moisture variation has a moderately effect on the natural frequencies of the FRC laminated beam, but only has a small effect on the nonlinear to linear frequency ratios of the same beam. In contrast, it has a significant effect on the nonlinear bending load–deflection curves of the FRC laminated beam. The results confirm that the thermal postbuckling equilibrium path of mid-plane unsymmetric FG-FRC laminated beams with immovable simply supported end conditions is no longer the bifurcation type.

Nonlinear analysis of functionally graded fiber reinforced composite laminated beams in hygrothermal environments, Part I: Theory and solutions

This paper investigates the large amplitude vibration, nonlinear bending and thermal postbuckling of anisotropic laminated beams resting on an elastic foundation in hygrothermal environments. The beam is made of fiber reinforced composites (FRCs) with the reinforcement being distributed either uniformly (UD) or functionally graded (FG) of piece-wise type along the thickness of the beam. The motion equations are based on a higher order shear deformation theory with a von Kármán-type of kinematic nonlinearity. The beam-foundation interaction and hygrothermal effects are also included, and the material properties of FRCs are estimated through a micromechanical model and are assumed to be temperature dependent and moisture dependent. A two-step perturbation technique is employed to determine the nonlinear to linear frequency ratios of beam vibration, the load–deflection curves of beam bending, and thermal postbuckling equilibrium paths of FRC laminated beams.

Thermal postbuckling of FGM cylindrical panels resting on elastic foundations

Thermal postbuckling analysis is presented for FGM cylindrical panels resting on elastic foundations. Heat conduction and temperature-dependent material properties are both taken into account. Material properties of functionally graded materials (FGMs) are assumed to be graded in the thickness direction based on Mori–Tanaka micromechanics model. The formulations are based on a higher order shear deformation theory and von Kármán strain displacement relationships. The panel–foundation interaction and thermal effects are also included. The governing equations are first deduced to a boundary layer type that includes nonlinear prebuckling deformation and initial geometric imperfection of the panel. Thermal postbuckling equilibrium paths are determined by using a singular perturbation technique along with a two-step perturbation approach. The effects of volume fraction index, foundation stiffness, panel curvature ratio on the thermal postbuckling behavior of FGM cylindrical panels are discussed in detail. The results reveal that the thermal postbuckling equilibrium path of FGM cylindrical panels with immovable simply supported edge conditions is no longer the bifurcation type for both uniform and non-uniform temperature variations.