Nonlinear Prebuckling Deformations Research Articles

Buckling and Postbuckling of Anisotropic Laminated Doubly Curved Panels under Lateral Pressure

This study presents buckling and postbuckling analyses of a thin anisotropic laminated doubly curved panel with rectangular planform subjected to lateral pressure. The material of each layer in the panel is assumed to be linearly elastic, anisotropic and fiber-reinforced. The governing equations are obtained by using classical shell theory and von Kármán strain displacement relations. Both nonlinear prebuckling deformations and initial geometric imperfections of the panel are taken into account. The closed-form solutions for buckling and postbuckling analyses of the thin anisotropic laminated doubly curved panel are performed using the two-step perturbation method and the boundary layer theory for shell buckling, which are employed to determine the interaction behavior of buckling loads and postbuckling equilibrium paths. Postbuckling response of perfect and imperfect, anisotropic laminated doubly curved panels with respect to the material and geometric properties and load-proportional parameters under different sets of thermal environmental conditions is numerically illustrated. New results illustrate that there exists the circumferential stress along with the associate shear stress and twisting when the anisotropic laminated doubly curved panel is subjected to lateral pressure. The shear stress, no matter how small it is, will affect the buckling loads and postbuckling equilibrium paths of the anisotropic doubly curved panels.

Radial postbuckling of nanoscaled shells embedded in elastic foundations based on Ru's surface stress elasticity theory

In this article, the nonlinear buckling and postbuckling response of cylindrical nanoshells under radial compressive load is investigated on the basis of a new surface elasticity theory. To this point, in contrast to the previous models based on the Gurtin-Murdoch elasticity theory, a new non-classical shell model based on the Ru's surface elasticity theory is developed in which the non-strain displacement gradient terms are eliminated from the surface stress-strain relations. The non-classical governing equations are derived using the principle of virtual work. Then they are deduced to boundary layer-type equations including simultaneously the nonlinear prebuckling deformations and the large postbuckling deflections. By using a singular perturbation technique, the new size-dependent problem is solved. It is indicated that due to the immovable edge supports and in the absence of the hydrostatic pressure, surface effects lead to an initial shortening caused by the induced residual strains. Additionally, the residual strain and initial surface tension terms vanish in the size-dependent nonlinear governing equations. Moreover, it is observed that this initial shortening diminishes quickly for thicker nanoshells.

Postbuckling of pressure-loaded FG-GRC laminated cylindrical panels resting on elastic foundations in thermal environments

This paper investigates the postbuckling behavior of graphene-reinforced composite (GRC) laminated cylindrical panels under lateral pressure loading. The panels are placed in a thermal environment and are supported by a Pasternak-type elastic foundation. The GRC layers are arranged in a piece-wise functionally graded (FG) pattern in the panel thickness direction. The anisotropic and temperature dependent material properties of GRC layers are predicted using the extended Halpin-Tsai micromechanical model. The higher order shear deformation theory and the von Karman strain-displacement relationships are employed to obtain the governing differential equations of the panels which also include the effects of temperature variation and panel-foundation interaction. The postbuckling of a cylindrical panel involves the boundary layer effect and the effects of the initial geometric imperfections. The nonlinear prebuckling deformations of the panel are also considered in the solution process. The postbuckling equilibrium path for the perfect and geometrically imperfect GRC laminated cylindrical panels are obtained by applying a singular perturbation technique along with a two-step perturbation approach. It is observed that, in the present examples, the considered FG-GRC laminated cylindrical panels subjected to lateral pressure experience pre-buckling deformation and the postbuckling equilibrium paths of the panels are no longer of the bifurcation type.

Postbuckling of doubly curved FG-GRC laminated panels subjected to lateral pressure in thermal environments

This article presents an investigation on the postbuckling behavior of doubly curved graphene-reinforced composite (GRC) laminated panels supported by an elastic foundation and subjected to lateral pressure and in thermal environments. The piece-wise GRC layers are arranged in a functionally graded (FG) pattern along the thickness direction of the panels. The overall mechanical properties of the FG-GRCs are assumed to be temperature dependent and are estimated through the extended Halpin-Tsai micromechanical model. The governing differential equations for the doubly curved panels are based on a higher order shear deformation shell theory with von Kármán strain-displacement relationships and the panel-foundation interaction. The initial deflections caused by lateral pressure and thermal bending stresses are both taken into account. The governing equations are first deduced to a boundary layer type that includes nonlinear prebuckling deformations and initial geometric imperfections of the panel. The postbuckling equilibrium path for the perfect and geometrically imperfect GRC laminated doubly curved panels are obtained by applying a singular perturbation technique along with a two-step perturbation approach. The impacts of material property gradient, temperature variation, panel curvature ratio, as well as foundation stiffness on the postbuckling behavior of FG-GRC laminated doubly curved panels are investigated.

Nonlinear torsional buckling and postbuckling analysis of cylindrical silicon nanoshells incorporating surface free energy effects

In the present study, a size-dependent shell model is developed which can afford to describe the nonlinear torsional buckling and postbuckling characteristics of cylindrical nanoshells in the presence of surface stress effects. To accomplish this purpose, the Gurtin–Murdoch theory of elasticity together with the von Karman geometric nonlinearity is implemented into the first-order shear deformation shell theory. A linear variation through the thickness is considered for the normal stress component of the bulk to satisfy the balance conditions on the free surfaces of the nanoshell. By means of the virtual work principle, the non-classical governing differential equations are constructed in which the transverse displacement and Airy stress function are considered as independent variables. Thereafter, a boundary layer theory is employed including the effect of surface stress in conjunction with the nonlinear prebuckling deformations and the large postbuckling deflections. Subsequently, an efficient solution methodology based on an improved perturbation technique is put to use to obtain the size-dependent critical torsional buckling loads and the associated postbuckling equilibrium paths. It is observed that the torsional load exhibits a significant increase after reaching the minimum postbuckling load. Also, it is revealed that the effect of surface stress becomes negligible at high values of the deflection.

Postbuckling behavior of shear deformable anisotropic laminated cylindrical shell under combined external pressure and axial compression

Structural design of composite shells are more challenging than conventional metals due to the complex mechanical behavior and damage mechanisms which composite materials exhibit. Postbuckling analysis for a moderately thick anisotropic laminated cylindrical shell subjected to combined loadings of external pressure and axial compression is presented which extends the boundary layer theory of shell buckling. The governing equations are based on Reddy’s higher order shear deformation shell theory with von Kármán-Donnell-type of kinematic nonlinearity. Both nonlinear prebuckling deformations and initial geometric imperfections of the shell are taken into account. A two-step singular perturbation method is used to determine interactive buckling loads and postbuckling equilibrium paths. A verification study is conducted, and the validity of the formulation is established through comparison with results of nonlinear finite element software such as ABAQUS®. The internal physical mechanism of the shell geometric parameters on the buckling load and the postbuckling equilibrium path is obtained. The numerical illustrations concern the postbuckling response of perfect and imperfect, moderately thick, anisotropic laminated cylindrical shells with different load-proportional parameters. The analytical model can provide an effective tool to investigate postbuckling of composite shell structures.

Nonlinear vibrations of pre- and post-buckled lipid supramolecular micro/nano-tubules via nonlocal strain gradient elasticity theory

The unique geometry with high surface ratio makes lipid micro/nano-tubules as an excellent self-assembled supramolecular structure in various biological applications such as controllable release systems and drug delivery. In the present study, the size-dependent nonlinear vibrations of axially loaded lipid micro/nano tubules associated with the both prebuckling and postbuckling domains are explored comprehensively. To accomplish this purpose, the nonlocal strain gradient theory of elasticity including simultaneously two entirely different features of size dependency is utilized within the framework of the third-order shear deformable beam model. With the aid of Hamilton's principle, the non-classical governing differential equations of motion are established incorporating the nonlinear prebuckling deformations and the large postbuckling deflections. At the end, the Galerkin method in conjunction with an improved perturbation technique is employed to initiate explicit analytical expressions for nonlocal strain gradient nonlinear frequency of pre- and post-buckled lipid micro/nano-tubules. It is seen that by taking the nonlocal size effect into consideration, the influence of geometrical parameters of the lipid micro/nano-tubule on the nonlinear vibration characteristics within the both prebuckling and postbuckling domains decreases and the frequency-deflection curves are more close to each other. However, the strain gradient size dependency has an opposite effect and leads to increase the gap between the frequency-deflection curves of axially compressed lipid micro/nano-tubules with different geometrical parameters.

Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells

Abstract The present study deals with the size-dependent nonlinear buckling and postbuckling characteristics of magneto-electro-elastic cylindrical composite nanoshells incorporating simultaneously the both of hardening-stiffness and softening-stiffness size effects. To accomplish this purpose, the nonlocal strain gradient elasticity theory is applied to the classical shell theory. Via the virtual work's principle, the size-dependent governing differential equations are constructed including the coupling terms between the axial mechanical compressive load, external magnetic potential and external electrical potential. The nonlinear prebuckling deformations and the large postbuckling deflections are taken into consideration based upon the boundary layer theory of shell buckling. Finally, an improved perturbation technique is employed to achieve explicit analytical expressions for nonlocal strain gradient stability curves of magneto-electro-elastic nanoshells under various surface electric and magnetic voltages. It is seen that a positive electric potential and a negative magnetic potential cause to increase both of the nonlocality and strain gradient size dependencies in the nonlinear instability behavior of axially loaded magneto-electro-elastic composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role.

Size-dependent axial instability of microtubules surrounded by cytoplasm of a living cell based on nonlocal strain gradient elasticity theory

Microtubules including tubulin heterodimers arranging in a parallel shape of cylindrical hollow plays an important role in the mechanical stiffness of a living cell. In the present study, the nonlocal strain gradient theory of elasticity including simultaneously the both nonlocality and strain gradient size dependency is put to use within the framework of a refined orthotropic shell theory with hyperbolic distribution of shear deformation to analyze the size-dependent buckling and postbuckling characteristics of microtubules embedded in cytoplasm under axial compressive load. The non-classical governing differential equations are deduced via boundary layer theory of shell buckling incorporating the nonlinear prebuckling deformation and microtubule-cytoplasm interaction in the living cell environment. Finally, with the aid of a two-stepped perturbation solution methodology, the explicit analytical expressions for nonlocal strain gradient stability paths of axially loaded microtubules are achieved. It is illustrated that by taking the nonlocal size effect into consideration, the critical buckling load of microtubule and its maximum deflection associated with the minimum postbuckling load decreases, while the strain gradient size dependency causes to increase them.

Thermo-electro-mechanical size-dependent postbuckling response of axially loaded piezoelectric shear deformable nanoshells via nonlocal elasticity theory

The current study addresses the size-dependent nonlinear thermo-electro-mechanical instability of piezoelectric nanoshells under a combined action of axial compressive load, lateral electric field and uniform changes in temperature. The non-classical formulations given herein are built upon the nonlocal elasticity theory within the framework of the shear deformation shell theory. On the basis of the minimum potential energy of the system and using a boundary layer theory of shell buckling, the nonlocal-based governing equations are deduced incorporating the nonlinear prebuckling deformations and initial geometric imperfection sensitivity. After that, a perturbation-based solution methodology is put to use in order to anticipate the size-dependent thermo-electro-mechanical postbuckling response of piezoelectric nanoshells corresponding to different nonlocal parameters, applied electric voltages and temperature changes. It is displayed that for the both local and nonlocal models with and without initial geometric imperfection, a lateral electric field coming from a positive applied voltage leads to increase the axial stiffness of piezoelectric nanoshell in such a way that the critical load and width of postbuckling domain increase, but no considerable change occurs in the minimum load of the postbuckling regime.

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.

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.

Surface free energy effects on the postbuckling behavior of cylindrical shear deformable nanoshells under combined axial and radial compressions

In the traditional continuum mechanics, the effects of surface free energy are generally ignored. However, this cannot be the case for nanostructures because of their high surface to volume ratio; surface energy plays an important role in the mechanical responses. In the present study, the nonlinear buckling and postbuckling characteristics of cylindrical nanoshells subjected to combined axial and radial compressions are investigated in the presence of surface energy effects. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical first-order shear deformation shell theory to develop an efficient size-dependent shell model incorporating surface free energy effects. Subsequently, a boundary layer theory is employed including surface effects in conjunction with the nonlinear prebuckling deformations, the large postbuckling deflections and the initial geometric imperfection. Finally, a solution methodology based on a two-stepped singular perturbation technique is utilized to obtain the size-dependent critical buckling loads and equilibrium postbuckling paths corresponding to the both axial dominated and radial dominated loading cases. It is observed that for the both axial dominated and radial dominated loading cases, surface free energy effects cause to increase the both critical buckling load and critical end-shortening of shear deformable nanoshell made of silicon.

Postbuckling of nanotube-reinforced composite cylindrical panels resting on elastic foundations subjected to lateral pressure in thermal environments

Modeling and analysis for the postbuckling of carbon nanotube-reinforced composite (CNTRC) cylindrical panels resting on elastic foundations subjected to lateral pressure in thermal environments are presented. Various profiles of single walled carbon nanotubes (SWCNTs) which are assumed to be uniformly distributed (UD) or functionally graded (FG) distribution along the thickness are taken into consideration. The temperature dependent material properties of FG-CNTRC panels are estimated through a micromechanical model. The formulations are developed based on a higher order shear deformation theory. To capture the large deflections, geometrical nonlinearity in von Kármán sense is taken into account. The panel-foundation interaction and thermal effects are also included. The initial deflections caused by lateral pressure and thermal bending stresses are both taken into account. The governing equations are first deduced to a boundary layer type that includes nonlinear prebuckling deformations and initial geometric imperfections of the panel. These equations are then solved by means of a singular perturbation technique along with a two-step perturbation approach. The influences of CNT volume fraction, temperature variation, panel geometric parameters as well as foundation stiffness on the postbuckling behavior of FG-CNTRC cylindrical panels are investigated.

Size-dependent buckling and postbuckling behavior of piezoelectric cylindrical nanoshells subjected to compression and electrical load

This paper studies the nonlinear buckling and postbuckling characteristics of piezoelectric cylindrical nanoshells subjected to an axial compressive mechanical load and an electrical load in the presence of surface free energy effects. The electrical field is applied along the transverse direction. A size-dependent shell model is adopted based on the Gurtin–Murdoch elasticity theory and von Karman geometrical nonlinearity. To satisfy the balance conditions on the surfaces of the nanoshell, a linear variation is considered for the normal stress of the bulk through the thickness. A boundary layer theory is employed including surface energy effects in conjunction with the effects of nonlinear prebuckling deformation, large deflections in the postbuckling regime and initial geometrical imperfections. Afterwards, a two-stepped singular perturbation technique is employed to obtain the size-dependent critical buckling load and the associated postbuckling equilibrium path for alternative electric loadings. It is found that the surface free energy and electrical load can cause an increase or decrease on the critical buckling load and the associated postbuckling strength of a nanoshell depending on the sign of surface properties and applied voltage. These anticipations are the same for the both perfect and imperfect piezoelectric nanoshells.

Size-dependent axial buckling and postbuckling characteristics of cylindrical nanoshells in different temperatures

One of the most important reasons that give rise to the extraordinary behaviors of nanostructures is the free surface energy. In the current investigation, a size-dependent shell model is introduced which has an excellent capability to take surface energy effects into account. To this end, Gurtin–Murdoch elasticity theory is implemented into the classical shell theory. Using virtual work׳s principle, the non-classical governing differential equations related to the cylindrical nanoshell subjected to axial compressive load are derived. Subsequently, a boundary layer theory is extended to solve the problem with considering the effects of surface free energy in addition to the nonlinear prebuckling deformations and the large postbuckling deflections. Finally, a solution methodology based on a two-stepped perturbation technique is put to use in order to obtain the size-dependent critical buckling loads and related postbuckling equilibrium paths corresponding to different surface properties and various sets of thermal environments. It is found that for all sets of thermal environment, the surface free energy has significant influence on the postbuckling strength of nanoshell. Also, it is seen that thermal environment causes to decrease the both critical buckling load and critical end-shortening, but it has a negligible influence on the value of minimum postbuckling load.

Postbuckling of pressure-loaded nanotube-reinforced composite doubly curved panels resting on elastic foundations in thermal environments

This paper presents an investigation on the postbuckling behavior of doubly curved nanocomposite panels reinforced by carbon nanotubes (CNTs) subjected to lateral pressure. The functionally graded carbon nanotube-reinforced composites (FG-CNTRCs) are assumed to have CNTs linearly graded in the thickness direction. The overall mechanical properties of the FG-CNTRCs, which include the thermal effect of CNTs and the matrix, are estimated through a micromechanical model. The panels may rest on elastic foundations. The governing differential equations for the doubly curved panels are based on a higher order shear deformation shell theory with von Kármán strain–displacement relationships and the panel–foundation interaction. The initial deflections caused by lateral pressure and thermal bending stresses are both taken into account. The governing equations are further deduced to a boundary layer type problem that includes nonlinear prebuckling deformations and initial geometric imperfections of the panels which are subsequently solved using a two-step perturbation approach. The influences of CNT volume fraction, temperature variation, panel geometric parameters, as well as foundation stiffness on the postbuckling behavior of FG-CNTRC doubly curved panels are investigated.

Postbuckling of pressure-loaded FGM doubly curved panels resting on elastic foundations in thermal environments

Modeling and analysis for the postbuckling of FGM doubly curved panels resting on elastic foundations and subjected to lateral pressure under heat conduction are presented. The initial deflections caused by lateral pressure and thermal bending stresses are both taken into account. The temperature-dependent 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 with 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 deformations and initial geometric imperfections of the panel. These equations are then solved by means of a singular perturbation technique along with a two-step perturbation approach. The effects of volume fraction index, temperature variation, the panel geometric parameters as well as foundation stiffness on the postbuckling behavior of FGM doubly curved panels are discussed in detail.

Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to axial compression

For structures at nanoscale, the surface effects can be important due to the high ratio of surface area to volume. In the current investigation, the nonlinear axial postbuckling behavior of geometrically imperfect cylindrical nanoshells is studied including surface stress effects. For this purpose, Gurtin–Murdoch continuum elasticity theory in conjunction with von Karman–Donnell-type geometric nonlinearity is implemented into the classical shell theory. By the developed size-dependent shell model, the surface effects which include surface elasticity and residual surface stress are taken into account. In order to satisfy balance conditions on the surfaces of nanoshell, a linear variation through the thickness is considered for the normal stress component of the bulk. Based on the variational approach using virtual work's principle, the non-classical governing differential equations are derived. In order to solve the nonlinear problem, a boundary layer theory is employed which contains simultaneously the nonlinear prebuckling deformations, initial geometric imperfections and large deflections corresponding to the postbuckling domain. Subsequently, a two-stepped singular perturbation methodology is utilized to predict the nonlinear critical buckling loads as well as the postbuckling equilibrium paths. It is observed that by taking surface stress effects into account, the both critical buckling load and critical end-shortening of a cylindrical nanoshell made of Silicon increase.

Surface stress effects on the nonlinear postbuckling characteristics of geometrically imperfect cylindrical nanoshells subjected to torsional load

The prime aim of the current investigation is to predict the nonlinear torsional buckling and postbuckling behavior of geometrically imperfect cylindrical nanoshells including surface stress effects. To this end, the size-dependent governing differential equations of cylindrical nanoshell based on von Karman–Donnell-type of kinematic nonlinearity are derived using a combination of Gurtin–Murdoch elasticity theory and the classical shell theory, and employing the principle of virtual work. Subsequently, by considering the transverse displacement and Airy stress function as independent variables, a boundary layer theory is put to use which takes surface stress effects into account in conjunction with the nonlinear prebuckling deformations, large postbuckling deflections and initial geometric imperfection. Finally, an efficient solution methodology based on a two-stepped singular perturbation technique is conducted to obtain the size-dependent postbuckling equilibrium paths of nanoshells. It is revealed that after the minimum point of postbuckling load, by increasing the value of applied torsional load, the difference between postbuckling load-deflection curves corresponding to the perfect and imperfect nanoshells tends to decrease.