Postbuckling Equilibrium Paths Research Articles

Imperfection sensitivity of the size-dependent nonlinear instability of axially loaded FGM nanopanels in thermal environments

The nonlinear buckling of thin shell-type structures is sensitive to the initial geometric imperfection. In this study, the imperfection sensitivity of the nonlinear instability of cylindrical nanopanels made of functionally graded material (FGM) is addressed including surface elasticity, aiming to present a background for axial postbuckling behavior of imperfect FGM nanopanels. The material properties are supposed to be graded across the panel thickness in accordance with a simple power law function of the volume fractions of the silicon and aluminum constituents with considering the physical neutral plane position. The non-classical governing differential equations are constructed and then they are deduced to boundary layer-type ones. Afterward, a perturbation-based solution methodology is employed to extract explicit expressions for the size-dependent postbuckling equilibrium paths of FGM nanopanels with and without initial geometric imperfection and corresponding to various panel thicknesses, geometrical parameters, temperature changes and material property gradient indexes. It is displayed that through reduction of the surface elasticity effects for thicker FGM nanopanels, the influence of the initial geometric imperfection on the minimum load of the postbuckling regime increases. This pattern is more significant for FGM nanopanels with a lower value of the material gradient index.

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.

Buckling and postbuckling of biaxially compressed functionally graded multilayer graphene nanoplatelet-reinforced polymer composite plates

The present paper investigates the biaxially compressed buckling and postbuckling behaviors of functionally graded multilayer composite plates reinforced with a low content of graphene nanoplatelets (GPLs) that are randomly oriented and uniformly dispersed in the polymer matrix within each individual layer. The material properties of the GPL-reinforced composite (GPLRC), which are graded along the thickness direction due to a layer-wise change in GPL weight fraction, are evaluated through a micromechanics model. Theoretical formulations are based on the first-order shear deformation plate theory and von Kármán-type nonlinear kinematics and include the effect of an initial geometric imperfection. A two step perturbation technique is employed to determine the asymptotic postbuckling solutions and the biaxial compressive postbuckling equilibrium paths of both perfect and imperfect plates simply supported on all edges. The effects of GPL weight fraction, distribution pattern, geometry and size as well as total number of layers on the buckling and postbuckling behaviors of functionally graded GPLRC plates are examined in detail.

Buckling and postbuckling of functionally graded graphene-reinforced composite laminated plates in thermal environments

Abstract Modeling and analysis of the postbuckling behavior of graphene-reinforced composite (GRC) laminated plates are presented in this paper. The GRC plates are in a thermal environment, subjected to uniaxial compression and resting on an elastic foundation. The temperature-dependent material properties of functionally graded graphene-reinforced composites (FG-GRCs) are assumed to be graded in the plate thickness direction with a piece-wise type, and are estimated through a micromechanical model. The postbuckling problem of FG-GRC laminated plates is modeled using a higher order shear deformation plate theory and the plate-foundation interaction and thermal effects are taken into consideration. A two-step perturbation technique is employed to determine the buckling loads and the postbuckling equilibrium paths. The compressive buckling and postbuckling behavior of perfect and imperfect, geometrically mid-plane symmetric FG-GRC laminated plates under different sets of thermal environmental conditions is obtained and is also compared with the behavior of uniformly distributed GRC laminated plates. The results show that the buckling loads as well as the postbuckling strength of the GRC laminated plates may be enhanced through piece-wise functionally graded distribution of graphene.

Thermal buckling and post-buckling analysis of functionally graded beams based on a general higher-order shear deformation theory

• Thermal buckling analysis of FGM beams by various theories are presented. • A two-step perturbation method is employed to determine the critical buckling loads and post-buckling equilibrium paths. • The post-buckling equilibrium path for FGM beam with two clamped ends is also of the bifurcation type for any various displacement fields. In the present work, attention is focused on the prediction of thermal buckling and post-buckling behaviors of functionally graded materials (FGM) beams based on Euler–Bernoulli, Timoshenko and various higher-order shear deformation beam theories. Two ends of the beam are assumed to be clamped and in-plane boundary conditions are immovable. The beam is subjected to uniform temperature rise and temperature dependency of the constituents is also taken into account. The governing equations are developed relative to neutral plane and mid-plane of the beam. A two-step perturbation method is employed to determine the critical buckling loads and post-buckling equilibrium paths. New results of thermal buckling and post-buckling analysis of the beams are presented and discussed in details , the numerical analysis shows that, for the case of uniform temperature rise loading, the post-buckling equilibrium path for FGM beam with two clamped ends is also of the bifurcation type for any arbitrary value of the power law index and any various displacement fields.

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.

Imperfection sensitivity of the size-dependent postbuckling response of pressurized FGM nanoshells in thermal environments

The purpose of the current study is to address the nonlinear buckling and postbuckling response of nanoscaled cylindrical shells made of functionally graded material (FGM) under hydrostatic pressure aiming to investigate the sensitivity to the initial geometric imperfection in the presence of surface effects and thermal environments. According to a power law distribution, the material properties of the FGM nanoshell are considered change through the shell thickness. Also, the change in the position of physical neutral plane corresponding to different volume fractions is taken into account to eliminate the stretching-bending coupling terms. In order to acquire the size effect qualitatively, the well-known Gurtin-Murdoch elasticity theory is incorporated within the framework of the classical shell theory. Using the variational approach, the non-classical governing equations are displayed and deduced to boundary layer type ones. Afterwards, explicit expressions for the size-dependent radial postbuckling equilibrium paths of imperfect FGM nanoshells are proposed with the aid of a perturbation-based solution methodology. It is displayed that by moving from the ceramic phase to the metal one, the critical buckling pressure decreases, but the postbuckling stiffness increases, because in contrast to the ceramic phase, the surface modulus and residual surface stress associated with the metal phase have the same sign.

Nonlinear vibration and postbuckling of functionally graded graphene reinforced porous nanocomposite beams

The nonlinear free vibration and postbuckling behaviors of multilayer functionally graded (FG) porous nanocomposite beams that are made of metal foams reinforced by graphene platelets (GPLs) are investigated in this paper. The internal pores and GPL nanofillers are uniformly dispersed within each layer but both porosity coefficient and GPL weight fraction change from layer to layer, resulting in position-dependent elastic moduli, mass density and Poisson's ratio along the beam thickness. The mechanical property of closed-cell cellular solids is employed to obtain the relationship between coefficients of porosity and mass density. The effective material properties of the nanocomposite are determined based on the Halpin-Tsai micromechanics model for Young's modulus and the rule of mixture for mass density and Poisson's ratio. Timoshenko beam theory and von Kármán type nonlinearity are used to establish the differential governing equations that are solved by Ritz method and a direct iterative algorithm to obtain the nonlinear vibration frequencies and postbuckling equilibrium paths of the beams with different end supports. Special attention is given to the effects of varying porosity coefficients and GPL's weight fraction, dispersion pattern, geometry and size on the nonlinear behavior of the porous nanocomposite beam. It is found that the addition of a small amount of GPLs can remarkably reinforce the stiffness of the beam, and its nonlinear vibration and postbuckling performance is significantly influenced by the distribution patterns of both internal pores and GPL nanofillers.

Nonlinear buckling and postbuckling behavior of cylindrical shear deformable nanoshells subjected to radial compression including surface free energy effects

The objective of the present investigation is to predict the nonlinear buckling and postbuckling characteristics of cylindrical shear deformable nanoshells with and without initial imperfection under hydrostatic pressure load in the presence of surface free energy effects. To this end, Gurtin-Murdoch elasticity theory is implemented into the first-order shear deformation shell theory to develop a size-dependent shell model which has an excellent capability to take surface free energy effects into account. A linear variation through the shell thickness is assumed for the normal stress component of the bulk to satisfy the equilibrium conditions on the surfaces of nanoshell. On the basis of variational approach and using von Karman-Donnell-type of kinematic nonlinearity, the non-classical governing differential equations are derived. Then a boundary layer theory of shell buckling is employed incorporating the effects of surface free energy in conjunction with nonlinear prebuckling deformations, large deflections in the postbuckling domain and initial geometric imperfection. Finally, an efficient solution methodology based on a two-stepped singular perturbation technique is put into use in order to obtain the critical buckling loads and postbuckling equilibrium paths corresponding to various geometric parameters. It is demonstrated that the surface free energy effects cause increases in both the critical buckling pressure and critical end-shortening of a nanoshell made of silicon.

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.

Numerical model of postbuckling behavior of GFRP beams subjected to pure bending

ABSTRACTThis article deals with numerical calculations of thin-walled composite laminate beams with closed cross-section subjected to pure bending. Thin-walled beams with square cross-section made of eight-layer GFRP laminate have been taken into consideration. An FEM model has been prepared using four-node multi-layered shell elements governed by first-order shear deformation theory. Linear buckling analysis and nonlinear static analysis including large displacements have been performed. A Newton–Rapson algorithm has been employed. The FEM calculations have been conducted using Ansys® software. Three following layer arrangement were considered: [45/−45/0/0]S, [45/−45/45/0]S and [45/−45/45/0/0/−45/45/−45]T. The influence of number of load steps and geometrical initial imperfection on course of equilibrium paths has been checked. The results of calculations have been compared with experimentally performed four-point bending tests. Comparing results of tests and numerical calculations, it was observed that, in some cases, the deflection of beams does not correspond to the bifurcation buckling mode but corresponds to the lowest buckling load; the geometrical imperfection that corresponds to the higher buckling mode should be taken into consideration to obtain similar postbuckling equilibrium paths from tests and FEM calculations. As is well known, the buckling load and failure load for real structures depend on course of equilibrium paths. Taking above considerations into account, the author of this article tries to show some problems in obtaining numerical calculations results that are similar to experimental tests.

FEM and EFG Quasi-Static Explicit Buckling Analysis for Thin-Walled Members

The quasi-static explicit finite element method (FEM) and element free Galerkin (EFG) method are applied to trace the post-buckling equilibrium path of thin-walled members in this paper. The factors that primarily control the explicit buckling solutions, such as the computation time, loading function and dynamic relaxation, are investigated and suggested for the buckling analysis of thin-walled members. Three examples of different buckling modes, namely snap-through, overall and local buckling, are studied based on the implicit FEM, quasi-static explicit FEM and EFG method via the commercial software LS-DYNA. The convergence rate and accuracy of the explicit methods are compared with the conventional implicit arc-length method. It is drawn that EFG quasi-static explicit buckling analysis presents the same accurate results as implicit finite element solution, but is without convergence problem and of less-consumption of computing time than FEM.

Size-Dependent Buckling and Postbuckling Analyses of First-Order Shear Deformable Magneto-Electro-Thermo Elastic Nanoplates Based on the Nonlocal Elasticity Theory

This paper presents a nonlocal nonlinear first-order shear deformable plate model for investigating the buckling and postbuckling of magneto-electro-thermo elastic (METE) nanoplates under magneto-electro-thermo-mechanical loadings. The nonlocal elasticity theory within the framework of the first-order shear deformation plate theory along with the von Kármán-type geometrical nonlinearity is used to derive the size-dependent nonlinear governing partial differential equations and associated boundary conditions, in which the effects of shear deformation, small scale parameter and thermo-electro-magneto-mechanical loadings are incorporated. The generalized differential quadrature (GDQ) method and pseudo arc-length continuation algorithm are used to determine the critical buckling loads and postbuckling equilibrium paths of the METE nanoplates with various boundary conditions. Finally, the influences of the nonlocal parameter, boundary conditions, temperature rise, external electric voltage and external magnetic potential on the critical buckling load and postbuckling response are studied.

Nonlocal postbuckling analysis of graphene sheets with initial imperfection based on first order shear deformation theory

In this paper, the first order shear deformation theory (FSDT) is used to investigate the postbuckling behavior of orthotropic single-layered graphene sheet (SLGS) under in-plane loadings. Nonlocal elasticity theory and von-Karman nonlinear model in combination with the isogeometric analysis (IGA) have been applied to study the postbuckling characteristics of SLGSs. In contrast to the classical model, the nonlocal continuum model developed in this work considers the size-effects on the postbuckling characteristics of SLGSs. FSDT takes into account effects of shear deformations through-the-thickness of plate. Geometric imperfection which is defined as a very small transverse displacement of the mid-plane is applied on undeformed nanoplate to create initial deviation in graphene sheet from being perfectly flat. Nonlinear governing equations of motion for SLGS are derived from the principle of virtual work and a variational formulation. At the end, the results are presented as the postbuckling equilibrium paths of SLGS. The influence of various parameters such as edge length, nonlocal parameter, compression ratio, boundary conditions and aspect ratio on the postbuckling path is investigated. The results of this work show the high accuracy of nonlocal FSDT-based analysis for postbuckling behavior of graphene sheets.

Size-dependent nonlinear instability of shear deformable cylindrical nanopanels subjected to axial compression in thermal environments

In this work, modeling and analysis for size-dependent buckling and postbuckling behavior of cylindrical nanopanels under axial compression in thermal environments are presented. To this end, a non-classical panel model based upon the Gurtin–Murdoch elasticity theory in conjunction with the first-order shear deformation shell theory is developed which considers efficiently the effects of surface free energy. In order to capture the large deflections, the von Karman geometrical nonlinearity is taken into account. The size-dependent governing differential equations are deduced to a boundary layer type that incorporates the nonlinear prebuckling deformations and the large postbuckling deflections. Afterwards, by means of an efficient solution methodology using a perturbation solving process, the size-dependent critical buckling loads and associated postbuckling equilibrium paths are obtained corresponding to various geometrical parameters and thermal environments. It is seen that for the both classical and non-classical panel models, the wide of snap-through phenomenon is larger for nanopanels with higher ratio of length to width. Also, it is found that thermal environment has no influence on the value of minimum load of the postbuckling domain and associated maximum deflection.

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.

Global and local interactive buckling behavior of a stiff film/compliant substrate system

This paper elucidates the global and local interactive buckling behavior of a stiff film resting on a compliant substrate under uniaxial compression. The resulting governing non-linear equations (non-autonomous fourth-order ordinary differential nonlinear equations with integral conditions) are then solved by introducing a continuation algorithm, which offers considerable advantages to detect multiple bifurcations and trace a complex post-buckling path. The critical conditions for local and global buckling and respective post-buckling equilibrium paths are carefully studied. Two different evolution mechanisms of buckling modes and processes from destabilization to restabilization (snap-back) are observed beyond the onset of the primary sinusoidal wrinkling mode in the post-buckling range. In addition, the shear modulus of an orthotropic substrate acts as a dominant role in the bifurcation portrait. Our results offer better understanding of the global and local buckling behaviors of such a bilayer system, and can open up new opportunities for the design and applications of novel nanoelectronics.

Thermal post-buckling of FG-CNT reinforced composite plates

Present research deals with the postbuckling problem of carbon nanotube reinforced composite plates subjected to uniform temperature rise loading. Distribution of carbon nanotubes as reinforcements may be uniform or functionally graded. To account for the large deformations of the plate, von-Kármán type of geometrical nonlinearity is included into the formulation. The virtual displacements principle associated with the conventional Ritz formulation whose shape functions are selected as the Chebyshev polynomials is used to obtain the matrix representation of the nonlinear equilibrium equations. The solution method is general and may be used for arbitrary combination of boundary conditions. The postbuckling equilibrium path which is governed by a nonlinear eigenvalue problem is traced using a displacement control strategy. Results of this study are compared with the available data in the open literature for the cases of isotropic homogeneous plates and cross-ply laminated plates. Afterwards numerical results are given for FG-CNTRC plates. It is shown that, FG-X pattern results in higher buckling temperature and also decreases the postbuckling deflection of the plate. Furthermore, this type of composites are eager to exhibit the secondary instability which is designated with a snap-through phenomenon in the post-buckling equilibrium path.

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.

Imperfection sensitivity of postbuckling behaviour of functionally graded carbon nanotube-reinforced composite beams

The imperfection sensitivity of the postbuckling behaviour of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) beams subjected to axial compression is investigated based on the first-order shear deformation beam theory with a von Kármán geometric nonlinearity. The material properties of FG-CNTRC are assumed to vary in the beam thickness direction and are estimated according to the extended rule of mixture. The differential quadrature method is employed to discretize the governing differential equations and the modified Newton-Raphson iterative technique is used to obtain the postbuckling equilibrium paths of FG-CNTRC beams with various imperfections. Parametric studies are carried out to examine the effects of imperfection modes, half-wave number, location, and amplitude on the postbuckling response of beams. The influences of CNT distribution pattern and volume fraction, boundary conditions, and slenderness ratio are also discussed. Numerical results in graphical form show that the postbuckling behaviour is highly sensitive to the imperfection amplitude. The imperfection mode and its half-wave number also moderately affect the imperfection sensitivity of the postbuckling response, whereas the effects of other parameters are much less pronounced.