Post-buckling Paths Research Articles

Nonlinear buckling and free vibration analysis of auxetic graphene origami composite beams under nonuniform thermal environment

This study examines the thermo-mechanical behavior of auxetic metamaterial beams enhanced by graphene origami (GOri) under spatially varying nonuniform temperature distributions (SVTD). Utilizing Timoshenko beam theory considering von-Kármánn type nonlinear strain–displacement relationship, GOri beams are modeled as layered structures. The Ritz method is employed to solve equilibrium equations, analyzing the impact of GOri distribution patterns, content, and folding degree on post-buckling and vibration paths. The effects of five SVTDs, three end conditions, and three GOri distribution patterns on buckling, post-buckling behavior, and nonlinear free vibration characteristics are explored. Findings reveal that the parabolic temperature distribution with peak temperatures at beam ends (P-MAE) results in higher critical temperatures and nonlinear free vibration frequencies. This research provides crucial insights into the design and optimization of GOri-enabled metamaterial structures in complex thermal environments, highlighting the significant influence of nonuniform temperature distributions along the beam’s length.

Thermal and thermo-mechanical post-buckling analysis of GNP-reinforced composite laminated plates

This research utilizes isogeometric analysis (IGA) method to analyze buckling and post-buckling responses of composite laminated plates embedded with graphene nanoplatelets (GNPs) subjected to three-dimensional (3D) conduction of heat or combined 3D heat conduction and mechanical edge compression. A formulation to determine temperature profile produced by 3D conduction of heat in the GNP-reinforced laminate is established, and a new function with smooth transition form of GNP volume fraction across the thickness is presented for dictating the volume fraction of layer. Shear deformable quasi-3D theory with the von Kármán type nonlinearity which takes initial geometric deformation and thermal effect into consideration is employed to construct the nonlinear equilibrium states. Four types of GNP arrangements encompassing uniform, O, X and V shapes are considered. The IGA approach presented, through the benchmark test, is evidenced to estimate the thermal buckling temperature accurately and to successfully trace the thermal post-buckling path. Additional parametric study is carried out to scrutinize the thermal and the thermo-mechanical post-buckling features of the GNP-reinforced composite laminated plates, and the new findings are sure to be served as the benchmark solutions.

Thermal Buckling and Postbuckling Analysis of Cracked FG-GPL RC Plates Using a Phase-Field Crack Model

A phase-field crack model is developed for numerical analysis of thermal buckling and postbuckling behavior of a functionally graded (FG) graphene platelet-reinforced composite (FG-GPLRC) plate with a central crack. The inclined central crack is represented according to a stable, effective phase-field formulation (PFF) by introducing a virtual crack rotation. The problem is formulated using first-order shear deformation theory (SDT) incorporated with von Kármán geometric nonlinearity. And it is approximated by combining regular Laplace interpolation functions and crack-tip singular functions in the framework of the 2D extended natural element method (XNEM). Troublesome shear locking is suppressed by applying the concept of the MITC (mixed-interpolated tensorial components)3+ shell element to the present numerical method. The results demonstrate the effectiveness of this method in accurately predicting the critical buckling temperature rise (CBTR) and the thermal postbuckling path. In addition, the parametric results reveal that the CBTR and postbuckling path of the FG-GPLRC plate are distinct from those of the FG carbon nanotube-reinforced composite (FG-CNTRC) plate and remarkably affected by the parameters associated with the crack and graphene platelet (GPL).

Thermal postbuckling and thermally induced postbuckled flutter of tri-directional functionally graded plates in yawed supersonic flow

The current work examines the thermal postbuckling, aero-elastic flutter and thermally induced postbuckled flutter about a static equilibrium state of tri-directional functionally graded (TFGM) rectangular and tapered plates in yawed supersonic flow. Based on the von Kármán nonlinear strains, the general higher-order shear deformation theory (GHSDT) and the first-order piston theory, the governing equations of motion for TFGM plates in yawed supersonic flow are established via the Hamilton's principle. The isogemetric analysis (IGA), the load continue strategy associated with the Newton Raphson iterative technique are exploited synthetically to capture the postbuckling paths, and then the postbuckled flutter behaviors are acquired with the static equilibrium state being obtained. Comparative and convergence studies are provided to improve reliabilities of the present material model, formulae and code implementations. Subsequently, detailed parametric investigations are carried out to evaluate the influences of material volume fractions, boundary conditions and airflow angles on the thermal postbuckling, aero-elastic flutter and postbuckled flutter behaviors of TFGM plates. The results show that it is the in-plane volume fractions, rather than the thickness volume fraction, that exert a greater influence on the thermal postbuckling and flutter behaviors of TFGM plates. Furthermore, TFGM plates exhibit more diverse postbuckling paths and more complex deformations due to the inhomogeneity of the in-plane material properties compared with z-directional FGM plates.

Three-dimensional elastoplastic post-buckling analysis of functionally graded plates using a novel meshfree Tchebychev-radial point interpolation approach

This paper presents a three-dimensional (3D) analysis of the post-buckling behavior of functionally graded (FG) plates for the first time using an elastoplasticity-based meshfree formulation. A novel 3D Tchebychev-radial point interpolation approach, which combines radial basis functions with Tchebychev polynomials, is developed to solve the nonlinear post-buckling problem. The incremental plastic deformation is modeled by the Prandtl–Reuss flow rule along the isotropic hardening von Mises criterion. The governing equations are derived using the principle of virtual work by considering the 3D full Green–Lagrange strain components. The Newton–Raphson technique along with the arc-length method is used to determine post-buckling equilibrium paths of FG plates. The effective elastoplastic properties of the functionally graded material are assessed by exploiting the homogenization method, named Tamura–Tomota–Ozawa (TTO) model. It has been demonstrated that the accuracy of the solution is improved by incorporating Tchebychev polynomials into the radial point interpolation method (RPIM) shape function, and the stability and robustness of the results are independent of variations in the shape parameter. Furthermore, it is confirmed that TRPIM, with a slightly higher CPU time compared to RPIM, exhibits a higher convergence rate. The excellent agreement of the results with those existing in the literature shows that the proposed meshfree method can be used as a robust and accurate numerical tool to predict the elastoplastic post-buckling behavior of FG plates. Further numerical assessments indicate that post-buckling paths are significantly affected by factors such as geometric parameters, material gradient, loading ratio, and boundary conditions (BCs).

Maximum buckling load design and post-buckling compressive failure of curved grid sub-stiffened composite panels

The structural efficiency of composite stiffened panels can be substantially improved by utilizing curved stiffening components due to their exceptional mechanical performance and design flexibility. This paper aims to explore the impressive potential offered by the curved stiffening concept in the design of T-shaped stiffened composite panels featuring curved sub-stiffening components. An optimization design framework is proposed to optimize the laminates sequence of skin, ply orientation angles and distribution of the curved sub-stiffeners as well as the curvature of the sub-stiffeners of the proposed curved grid sub-stiffened composite panels to yield higher buckling performance without additional weight. Results demonstrate that introducing composite curved grid sub-stiffeners to the T-shaped stiffened composite panel leads to substantial improvements (+254.42%) in buckling performance, while maintaining equivalent weight designs. Furthermore, the introduction of composite curved grid sub-stiffeners reveals alterations to the post-buckling deformation paths of grid sub-stiffened composite panels, leading to varied collapse loads and degrees of deformation in the composite panels. This highlights the importance of utilizing the composite curved grid sub-stiffeners to improve stability and reduce the risk of failure in thin-walled structures, all while meeting lightweight design criteria.

Three-dimensional elastoplastic post-buckling analysis of porous FG plates resting on Winkler/Pasternak foundation using meshless RRKPM

This paper presents a three-dimensional (3D) analysis of the elastoplastic post-buckling behavior of porous functionally graded (FG) plates resting on Winkler/Pasternak foundations under uniaxial and biaxial in-plane loadings using an enhanced meshless approach. The principle of virtual work is used to derive the governing equations, and 3D nonlinear Green–Lagrange strains are considered. Plastic deformation is modeled employing the Prandtl–Reuss flow law and the isotropic hardening von Mises criterion. A novel meshless radial basis reproducing kernel particle approach is used to obtain the discretized system of equations. The Newton–Raphson method, coupled with the arc-length technique, is used to compute post-buckling paths of porous FG plates. Numerical assessments show that post-buckling paths are significantly influenced by porosity parameter, porosity distribution, foundation parameters, material gradient, plate thickness, loading ratio, and boundary conditions (BCs).

An insight into the mechanical behavior and failure mechanisms of T-stiffened composite structures with through-interface debonding defects

A robust 3D finite element model framework was established to capture the mechanical behavior and failure mechanisms of intact and through-interface debonding T-stiffened composite structures based on the progressive damage model and the cohesive zone model. Meanwhile, specimens with and without interface debonding defects were fabricated to validate the reliability and robustness of the proposed modeling framework. Finally, a parametric study was conducted on the types of interfacial debonding to further explore the effect on the buckling load, delamination initiation load, and ultimate load-carrying capacity of the composite stiffened structure. The results indicate that the load-carrying capacity of both single interface defects and double interface defects decreases gradually with increasing defect length and converges to a constant value. The reductions were 28.31% and 47.68%, respectively, compared to the intact structure. Interestingly, interfacial debonding has the most significant effect on the ultimate load at the 1/4 position. Moreover, interface debonding defects not only severely weaken the ultimate load-carrying capacity of the structure but also significantly alter the post-buckling path and failure mechanism, resulting in distinct degrees of deformation and damage morphology.

Experimental study of energy‑absorbing and support characteristics of glass microsphere-filled steel tube columns under uniaxial compression.

An innovative energy-absorbing and bearing structure was proposed, which incorporated the coupling of glass microspheres with a metal tube. Glass microsphere-filled steel tube (GMFST) column, consisting of external steel tube and inner glass microspheres, was expected to give full play to the energy-absorbing and load-bearing capacities of the particle while restricting particle flow from collapsing, thereby enhancing the overall structural strength. Four groups of steel tubes and the GMFST specimens were designed and subjected to axial compression tests at four different loading rates to investigate the performance of the structure. These tests aimed to analyze the deformation mode, mechanical response, and energy absorption capacity of the GMFST columns under quasi-static to low-speed compression conditions. The results indicated that the deformation process and failure mode of GMFST columns were similar to those of hollow steel tubes, albeit with a different post-buckling mode. Filling the steel tubes with glass microspheres reduced the load fluctuation range, moderated load-displacement curves, and exhibited a strain rate strengthening effect. The GMFST columns demonstrated superior energy absorption capacity, with significant increases in crush force efficiency, the averaged crush force, and the total absorbed energy, particularly in terms of subsequent support capacity. The load-increasing reinforcement properties enabled GMFST columns to overcome the limitations associated with the unstable post-buckling path of energy‑absorbing damping structure, exhibiting outstanding load-bearing performance and stability in the later stages. The results provided valuable guidelines for designing and engineering high-performance GMFST columns, serving as a new type of energy-absorbing and supporting structure.

Buckling and post-buckling analysis of double-layer magnetoelectric nano-plate strips considering piezo-flexoelectric and piezo-flexomagnetic effects

In the present work, closed-form solutions for the linear and nonlinear stability of the magnetoelectric nanoplate strips consisting of piezo-flexoelectric and piezo-flexomagnetic layers are presented. In the framework of the nonlocal strain gradient and first-order shear deformation theories, magneto-electro-mechanical responses are obtained for simply supported and clamped end conditions. Excellent agreements are observed between the obtained results and the existing literature. The effect of several parameters on critical buckling load and post-buckling path is investigated. Our outcomes reveal that the post-buckling load-carrying capacity increases considerably in the case of a double-layer ME nano-plate strip compared to a single-layer (piezo-flexoelectric and piezo-flexomagnetic) with the same thickness and length. Further, the post-buckling deformation decreases with a decrease in initial electric voltage and an increase in initial magnetic potential. The closed-form solutions and numerical results of this work are useful for future analyses and the design of such nanostructures.

Thermo-electrical postbuckling behavior of carbon nanotubes-reinforced composite beams with piezoelectric layers and tangentially restrained ends

An analytical investigation on the buckling and postbuckling behavior of carbon nanotube-reinforced composite beams integrated with surface-bonded piezoelectric layers under uniform temperature rise is presented in this paper. Carbon nanotubes (CNTs) are reinforced into isotropic matrix through uniform distribution and functionally graded distributions. The properties of material constituents are assumed to be temperature-dependent and effective properties of CNT-reinforced composite are estimated using an extended rule of mixture. Equilibrium equations of the beams are established based on Euler-Bernoulli theory including von Kármán nonlinearity and solved using analytical solutions and Galerkin method. Critical temperatures and postbuckling load-deflection paths are determined using an iteration algorithm. Parametric studies are performed to examine the influences of CNT distribution and volume fraction, applied voltage, in-plane and out-of-plane conditions of the ends, slenderness, and thickness ratio of layers on the critical loads and postbuckling load carrying capacity of beams. Results reveal that CNT volume fraction and degree of in-plane ends constraint have slight and significant influences on the critical temperatures and thermal postbuckling paths, respectively. The study also finds that negative and positive voltages increase and decrease the thermal buckling temperatures of piezoelectric CNT-reinforced composite beams.

Exact solution of thermal postbuckling of porous piezoelectric nanobeams with surface effect

In this paper, the thermal postbuckling of porous piezoelectric nanobeams is analyzed using the first-order shear deformation beam theory based on the surface elasticity theory and "core-shell" model, where the surface effect is introduced by employing the surface energy model. The governing equations for thermal postbuckling of porous piezoelectric nanobeams were derived by the principle of minimum potential energy. It is found that surface effect increases the effective elastic modulus and dimensionless critical load and reduces the thermal postbuckling conformations, thermal postbuckling paths and amount of induced charge. Meanwhile, the reasonable pore distribution pattern can effectively improve the mechanical properties.

Postbuckling of Laminated Composite Circular Conical Shells Under Combined Thermo-Mechanical Loads

In this paper, the postbuckling characteristics of cross- and angle-ply laminated composite conical shells subjected to combined thermo-mechanical loading are studied using shear deformable semi-analytical finite element. The nonlinear governing equations, considering geometric nonlinearity based on Sanders type of kinematic approximations, are solved using Newton-Raphson iteration procedure coupled with adaptive displacement control method to trace the postbuckling equilibrium path. It is brought out from the study that the inward displacement in the postbuckling region is significantly higher as compared to the outward one. The ratio of the lowest load in the postbuckling path and the bifurcation load increases with the increase in the pressure loading proportion and decreases with the increase in the axial loading proportions. The relative load carrying capacity of shells under different set of loading combinations changes with the increase in the maximum postbuckling displacement. It is also observed that the nonlinearity in the load interaction relation corresponding to lowest load in the postbuckling path is less compared to that for bifurcation load.

Influence of surface effect on post-buckling behavior of piezoelectric nanobeams

Piezoelectric nanobeams with excellent mechanical, thermal and electrical properties are important components in micro-nano electromechanical systems, which are widely used as sensors, brakes and resonators. Based on the Euler–Bernoulli beam model, the influence of surface effect on the post-buckling behaviour of piezoelectric nanobeams is analysed. According to the surface elasticity theory and the ‘core–shell’ model, the surface energy model is used to introduce the influence of surface effect. The governing equations and boundary conditions of the post-buckling of piezoelectric nanobeams under the influence of surface effect are derived by the principle of minimum potential energy. The analytical solution of post-buckling is obtained by the eigenvalue method. The influence of surface effect on the post-buckling configuration, post-buckling path, amount of induced charge and critical load of piezoelectric nanobeams with different external constraints and cross-sectional dimensions are discussed. The results show that surface effect has a significant influence on the post-buckling of piezoelectric nanobeams. Considering surface effect, the effective elastic modulus and critical load of piezoelectric nanobeams are increased, and the post-buckling configuration, post-buckling path and amount of induced charge are reduced. These findings contribute to the study of micro-nano electromechanical systems based on nanobeam structures and provide a theoretical basis for the design and manufacture of nanodevices.

Buckling optimization and post-buckling analysis of omega sub-stiffened composite panels using different cohesive interface properties

This paper investigates the effect of a new sub-structural configuration (omega sub-stiffener) on the buckling and post-buckling performance of an existing T-stiffened composite panel design. An optimization framework is proposed to optimize the distribution of omega sub-stiffeners, the laminate stacking sequences, and the ply orientation angles of omega sub-stiffened composite panels to yield higher initial buckling performance without adding mass. Further, different stiffener-panel cohesive interface properties are considered to analyze the underlying mechanism of omega sub-stiffeners involved in the panel’s post-buckling deformation paths and ultimate failure load during the compressive post-buckling process. Results show that introducing omega sub-stiffeners into the T-stiffened composite panel can result in an increased initial buckling performance (+242.63%) and causes re-establishment of the concentrated stress region, which yields different degrees of deformation and collapse in composite panels, and alters the failure behavior and further affects the initiation and propagation of various failure modes.

Buckling and Post-Buckling of Bidirectional Porous Beam Under Bidirectional Hygrothermal Environment

In this paper, buckling and nonlinear post-buckling behaviors of a bidirectional porous (BDP) beam are investigated under bidirectional hygrothermal environment. Euler–Bernoulli beam theory with the von Kármán nonlinearity is employed to derive the nonlinear variable coefficient governing differential equations based on Rayleigh quotient method. Analytical solutions of critical buckling load and load–deflection equilibrium path in post-buckling are deduced for the single directional varying (SDV) porous beam. The general numerical solutions for bidirectional varying (BDV) porous beam are obtained by differential quadrature finite element method (DQFEM) with Newton–Raphson iteration method based on the variation principle. The high accuracy of the present numerical method with higher computing efficiency is verified by comparison with published reports and the analytical results in this work. Parametric analysis on effects of the porosity bidirectional distributions, porosity coefficients, distributions of hygrothermal environment and boundary conditions on buckling load and post-buckling response is carried out to enhance the buckling and deformation resistances in design, manufacture and usage of porous structures. The results show that the bidirectional porosity pattern, linear and nonlinear hygrothermal distribution and boundary conditions play a significant role on buckling critical external load and critical hygrothermal increments, buckling form and post-buckling path.

The Influence of GPL Reinforcements on the Post-Buckling Behavior of FG Porous Rings Subjected to an External Pressure

The work focuses on the post- buckling behavior of functionally graded graphene platelet (FG-GPL)-reinforced porous thick rings with open-cell internal cavities under a uniform external pressure. The generalized rule of mixture and the modified Halpin–Tsai model are here used to evaluate the effective mechanical properties of the ring. Three types of porosity patterns are assumed together with five different GPL distributions as reinforcement across the ring thickness. The theoretical formulation relies on a 2D-plane stress linear elasticity theory and Green strain field in conjunction a virtual work principle to derive the nonlinear governing equations of the post-buckling problem. Unlike the simple ring models, 2D elasticity considers the thickness stretching. The finite element model combined with an iterative Newton–Raphson algorithm is used to obtain the post-buckling path of the ring up to the collapse. A systematic investigation evaluates the effect of the weight fraction of nanofillers, the coefficient of porosity, porosity distribution, and the GPLs distribution on the deep post-buckling path of the ring. Based on the results, it is found that the buckling value and post-buckling strength increase considerably (by approximately 80%) by increasing the weight fraction of the nanofiller of about 1%.

Stability Analyses of Cracked Functionally Graded Graphene-Platelets Reinforced Composite Beam Covered with Piezoelectric Layers

As cracks are unavoidable and always reduce structural local stiffness and strength, this paper pays attention to the effect of cracks on the stability of the cracked functionally graded (FG) graphene-nanoplates reinforced composite (GRC) beam covered with piezoelectric layers. Both the critical buckling loads and postbuckling paths of the novel structures with cracks are considered. The massless rotational spring model is employed to calculate the bending stiffness of the cracked section. Three different graphene platelets (GPLs) distribution patterns along the thickness direction of the FG-GRC core beam are studied. The effective material properties of the FG-GRC core beam are calculated by Halpin–Tsai model and the rule of mixture. The governing equations of stability of the cracked FG-GRC piezoelectric beam are established within the framework of the first-order shear deformation beam theory, von Kármán geometric nonlinearity and Ritz method. The direct iteration method is used to examine the effects of boundary conditions, crack parameters, piezoelectric layers and GPL parameters on the critical buckling loads and postbuckling responses of the cracked FG-GRC piezoelectric beams. Results clearly illustrate that GPLs can significantly improve the stability of the cracked FG-GRC piezoelectric beams, while the increasing crack depth has the opposite effect.

Postbuckled vibration behaviour of skew sandwich plates with metal foam core under arbitrary edge compressive loads using isogeometric approach

In the present work, nonuniform rational B-spline (NURBS) based isogeometric formulation in conjunction with refined higher-order theory is used to investigate the linear buckling, post-buckling, and post-buckled vibration behaviour of initially imperfect skew sandwich plates. The face sheets are functionally graded carbon nanotube-reinforced composite (FGCNTRC), and the core layer is made up of aluminium foam. The effects of three types of CNT distributions (uniform, FGX and FGO) in the face sheets, two types (uniform, symmetric) of porosity distribution functions for the core layer and five types of in-plane compressive loads are examined in the present investigation. The pre-buckling stresses are calculated using static analysis to evaluate accurate, critical loads. The post-buckling paths are traced using the modified Riks method. The accuracy of the present results is ascertained by comparing the results for critical loads and post-buckling paths with those available in the literature. Subsequently, the influence of CNT distribution functions, porosity functions, compressive loads, skew angle and the side-to-thickness ratio is studied on the nonlinear stability and free vibration behaviour of the post-buckled skew sandwich plates. The obtained results highlight that the buckling strength can be improved by increasing the skew angle, increasing the concentration of CNTs towards the surface of the face sheets and by using a metal foam core with nonuniform porosity distribution.

Three-dimensional postbuckling analysis of thick hyperelastic tubes

While the buckling of tubes under axial compression has been extensively studied, the postbuckling behavior of thick tubes remains elusive. In this paper, we conduct three-dimensional buckling and postbuckling analysis for thick hyperelastic tubes subjected to axial compression under finite deformation by the asymptotic expansion method. Our theoretical results successfully predict the deformation and stress-strain curves of buckled tubes near the critical loading, which are well validated by finite element analysis. Depending on the geometry, three kinds of postbuckling paths, including continuous buckling, snap-through and snap-back, are discovered. We summarize our results in two phase diagrams of the critical stretch for the onset of buckling and postbuckling paths with respect to the geometric parameters. In particular, we have observed that the postbuckling response can undergo a complex transition among different types of postbuckling paths, including continuous buckling, snap-through and snap-back, which is attributed to the competition between two modes of deformation, i.e., global deformation and local distortion. When a tube is long and thick, it prefers global deformation, and its cross section remains almost a plane after buckling, whereas when a tube is relatively short and relatively thin, it prefers local distortion and its cross section does not remain a plane any more after buckling. Our work provides understanding and insights into the buckling and postbuckling of thick tubes, and bridges the knowledge gap between postbuckling of thick columns and tubes.