Nonlinear Postbuckling Analysis Research Articles

Analysis of crease-wrinkle interaction for thin sheets

In this paper, geometrically non-linear post-buckling analyses were performed to study the effect of sheet thickness, deployment angle, and load ratio on the crease-wrinkle interaction. A square sheet configuration with a single transverse crease was modeled using thin shell elements. The analysis proceeded by initially providing a realistic deployed state of a creased membrane sheet. Then an uneven corner loading was applied to introduce wrinkling. The effects of the induced anisotropy from the crease on the fine-scale detail of the wrinkle evolution, as a function of sheet thickness, loading, and crease deployment angle were systematically investigated. Significant differences were found in sheet compliance and crease-wrinkle interaction as these parameters were varied.

Stretch-induced stress patterns and wrinkles in hyperelastic thin sheets

Wrinkles are commonly observed in stretched thin sheets and membranes. This paper presents a numerical study on stretch-induced wrinkling of hyperelastic thin sheets based on nonlinear finite element analyses. The model problem is set up for uniaxial stretching of a rectangular sheet with two clamped ends and two free edges. A two-dimensional stress analysis is performed first under the plane-stress condition to determine stretch-induced stress distribution patterns in the elastic sheets, assuming no wrinkles. As a prerequisite for wrinkling, development of compressive stresses in the transverse direction is found to depend on both the length-to-width aspect ratio of the sheet and the applied tensile strain in the longitudinal direction. A phase diagram is constructed with four different distribution patterns of the stretch-induced compressive stresses, spanning a wide range of aspect ratio and tensile strain. Next, an eigenvalue analysis is performed to find the potential buckling modes of the elastic sheet under the prescribed boundary conditions. Finally, a nonlinear post-buckling analysis is performed to show evolution of stretch-induced wrinkles. In addition to the aspect ratio and tensile strain, it is found that the critical condition for wrinkling and the post-buckling behavior both depend sensitively on the sheet thickness. In general, wrinkles form only when both the magnitude and the distribution area of the compressive stresses are sufficiently large. The wrinkle wavelength decreases with increasing strain, in good agreement with the prediction by a scaling analysis. However, as the tensile strain increases, the wrinkle amplitude first increases and then decreases, eventually flattened beyond a moderately large critical strain, in contrast to the scaling analysis.

Nonlinear Thermomechanical Post-Buckling Analysis of Thin Functionally Graded Annular Plates Based on Von-Karman's Plate Theory

In this article, a nonlinear bending and axisymmetric thermal buckling analysis of thin functionally graded (FG) annular plates is presented based on classical nonlinear Von-Karman's plate theory. Using a shooting method the nonlinear ordinary differential equations are solved numerically. The presented solution is validated by comparing the obtained results with those of an isotropic plate. The emphasis is placed on investigating the effect of varying the gradient index of the FG plate and thermal boundary conditions on the nonlinear bending and post buckling behavior of the FG plate in detail.

Large Deflection Behavior of Composite Laminated Plates under End-Shortening and Normal Pressure Loading Using a Semi-Energy Finite Strip Method

In this paper, the application of previously the semi energy finite strip method (FSM) for the non-linear post-buckling analysis of rectangular anti-symmetric laminates is extended to include the effects of normal pressure loading in addition to the progressive end-shortening. One of the main advantages of the semi-energy FSM is that it is based on the closed form solution of von Kármán’s compatibility equation. The developed finite strip method is applied to analyze the large deflection behavior of anti-symmetric angle ply composite laminated plates with simply supported boundary conditions at its loaded ends. To validate the results, they are compared with those obtained from finite element method (FEM) of analysis.

A nonlinear double-superposition global–local theory for dynamic buckling of imperfect viscoelastic composite/sandwich plates: A hierarchical constitutive model

Abstract Nonlinear dynamic thermo-mechanical buckling and postbuckling analyses of imperfect viscoelastic composite laminated/sandwich plates are performed by a proposed theory that takes into account all the interlaminar kinematic and transverse stress continuity conditions, for the first time. Even the dynamic buckling analysis of the multi-layered/sandwich plates employing the hierarchical constitutive model has not been performed before. The proposed theory is a double-superposition high-order global–local theory that is calibrated based on the nonlinear strain–displacement expressions for the thermoelastic loadings taking into account the structural damping. The buckling loads are determined based on a criterion previously published by the author. Various complex sensitivity analyses evaluating effects of the relaxation parameters, rate of the loading, sudden heating, and pre-stress on thermo-mechanical buckling of the viscoelatic multi-layered/sandwich plates are performed. Results show that the viscoelastic behavior may decrease the buckling load. Sudden dynamic buckling loads are higher due to the reflected stress waves.

접힌자국이 있는 멤브레인에서 두께에 따른 주름거동의 변화

본 논문에서는 코너에서 대각선방향으로 인장하중을 받고 있는 접힌자국이 있는 사각형 멤브레인에서 두께가 주름 거동에 미치는 영향을 기하학적 비선형 후좌굴 유한요소해석을 통하여 연구하였다. 멤브레인은 쉘 요소로 모델링 하였고 좌굴을 발생시키기 위하여 면외방향으로 미소의 무작위성 기하학적 결함을 메쉬에 가하였다. 해석은 접히지 않은 원멤브레인과 수직방향으로 접힌 멤브레인에 대해 수행하였고 그 결과를 비교하였다. 해석결과 멤브레인의 두께가 감소함에 따라 주름의 발생과 성장을 크게 촉진함을 알 수 있었다. 또한 접힌 자국의 초기 전개각이 증가할수록 국부주름의 낮은 하중에서 발생하였으나 전역주름으로의 성장은 지연되는 경향을 보였다. In this paper, the thickness effect on the wrinkle-crease interaction behavior of corner-loaded creased square membranes was studied using geometrically nonlinear post-buckling analysis. The membranes were modeled using shell elements, and the meshes were seeded with semi-random geometrical imperfection to instigate the buckling deformation. Results indicated that the wrinkle-crease interaction behavior was significantly dependent on the membrane thickness. Both the global and local wrinkles developed earlier as the thickness decreased. It was also found that the wrinkling behavior depended on the initial deployment angle in which the local wrinkle initiation occurred earlier, while the global wrinkle formation was delayed as the angle increased.

접힌 자국이 있는 멤브레인의 주름 거동 해석

본 논문에서는 수직방향으로 접은 사각형 멤브레인의 주름 거동을 기하학적 비선형 후좌굴 해석을 사용하여 연구하였다. 멤브레인은 쉘 요소로 모델링 하였고 하중은 멤브레인의 코너에서 대각선 방향으로 가하였다. 해석에는 다양한 각도의 초기 전개각을 가지는 멤브레인을 고려하였고 접힌 자국이 없는 경우에 대해서도 해석을 수행하여 그 결과를 비교하였다. 해석결과 주름은 큰 코너하중이 가해진 지역에서 국부적으로 발생하였는데, 이 국부 주름은 하중 비의 증가에 따라 점차 성장하여 전역 주름으로 발전하였다. 또한 주름의 발생 및 성장 거동은 접힌 자국의 초기 전개각에 좌우되는 경향을 보였다. In this paper, the wrinkling behavior of vertically creased corner-loaded square membranes was studied using geometrically nonlinear post-buckling analysis. The membranes were modeled using shell elements, and the meshes were seeded with semi-random geometrical imperfection to instigate the buckling deformation. A pristine and creased membranes with various initial deployment angles were considered in the analyses and the results were compared. Results showed that local wrinkles initiated near the corner where the higher load was applied, which grew to form a single diagonal global wrinkle as the load ratio increased. It was also found that the local wrinkle initiation and the global wrinkle formation were significantly dependent on the initial deployment angles.

Stability analysis of shells of revolution under pressure conditions

The subject of the present paper is shells of revolution subjected to the external pressure. A family of shells of revolution, with positive and negative Gaussian curvature, were taken into consideration. The volume V s and length L of the shell were constant. Three different r c / t ratios were investigated: 300, 504 and 700. The goal is to investigate elastic stability of shells of revolution. For that reason two kinds of analyses were performed using ABAQUS system. These are linear eigenvalue buckling prediction and non-linear post-buckling analysis. As a result of the linear analysis the influence of a meridional radius of curvature on the critical load was determined and presented on the graph. Non-linear analysis allowed to create equilibrium paths showing the behaviour of shells in a post-critical state. As can be seen from a number of plots convex barrelled shells are not stable after exceeding the critical load. Some concave barrelled shells on the contrary have stable equilibrium paths in the post-critical range.

Buckling of a stiff film bound to a compliant substrate—Part I:: Formulation, linear stability of cylindrical patterns, secondary bifurcations

The buckling of a thin elastic film bound to a compliant substrate is studied: we analyze the different patterns that arise as a function of the biaxial residual compressive stress in the film. We first clarify the boundary conditions to be used at the interface between film and substrate. We carry out the linear stability analysis of the classical pattern made of straight stripes, and point out secondary instabilities leading to the formation of undulating stripes, varicose, checkerboard or hexagonal patterns. Straight stripes are found to be stable in a narrow window of load parameters only. We present a weakly nonlinear post-buckling analysis of these patterns: for equi-biaxial residual compression, straight wrinkles are never stable and square checkerboard patterns are found to be optimal just above threshold; for anisotropic residual compression, straight wrinkles are present above a primary threshold and soon become unstable with respect to undulating stripes. These results account for many of the previously published experimental or numerical results on this geometry.

A continuum mechanics nonlinear postbuckling analysis for single-walled carbon nanotubes under torque

Abstract Nonlinear buckling and postbuckling analysis of single-walled carbon nanotubes subjected to torsional load are examined by continuum mechanics elastic shell model. The boundary layer theory of shell buckling and the singular perturbation technique is applied to solve the governing equations based on the Karman–Donnell-type nonlinear differential equations. The buckling loads and postbuckling equilibrium paths are obtained theoretically. For more detailed descriptions on the postbuckling behaviors of single-walled carbon nanotubes, postbuckling relations between twist, end-shortening, transverse deflection and the torque are investigated for nanotubes with different geometric parameters. Numerical results obtained show that single-walled carbon nanotubes under the torsional load have unstable postbuckling behavior, and the postbuckling equilibrium path is higher for the thinner nanotube than that of the thicker nanotubes. Furthermore, a new conclusion drawn from the postbuckling analysis on single-walled carbon nanotubes with different Young's modulus and the wall thickness is significant to definite the mechanical parameters of single-walled carbon nanotubes.

Post buckling analysis of smart laminated doubly curved shells

Abstract In the present investigation, the geometrically nonlinear post buckling analysis of piezoelectric laminated doubly curved shells is presented using finite element method. The piezoelectric material is used in the form of layers or patches embedded and/or surface bonded on laminated composite shells. The finite element model includes the general geometric nonlinearity due to large deflection. The shell geometry used in the formulation is derived using the orthogonal curvilinear coordinate system. Based on the principle of virtual work the nonlinear finite element equations are derived. A total Lagrangian approach associated to arc-length method is used to solve the equilibrium equations. Smart shells having integrated piezoelectric actuators undergo large displacements resulting snap through phenomenon from one equilibrium state to another. The present results are found to compare well with those available in the open literature and the post buckling responses of [±45°/∓45°] s laminated spherical, cylindrical and conoidal shell panels with piezoelectric layer are analyzed and the nonlinear load–deflection curves are presented.

Post buckling analysis of laminated composite shells

In the present investigation the geometrically nonlinear post buckling analysis of laminated composite doubly curved shells is presented using finite element method. The finite element model includes the general geometric nonlinearity due to large deflection. The present nonlinear strain displacement relations are expressed in the curvilinear coordinates. The material behaviour is, however, assumed to be linear and elastic. The principle of virtual work forms the basis to derive the nonlinear finite element equations. To solve the nonlinear finite element equations an incremental iterative technique based on the arc length method is employed. The present results are found to compare well with those available in the literature and are also able to capture both snap through and snap back post buckling behaviour. The nonlinear responses of three specially laminated spherical, cylindrical and conoidal shells are analyzed and the results are discussed.

Finite Element Analysis of Debonded Sandwich Beams Under Axial Compression

A nonlinear finite element analysis was performed to simulate axial compression of sandwich beams with debonded face sheets. The load-end-shortening diagrams were generated for a variety of specimens used in a previous experimental study. The energy release rate at the crack tip was computed using the J-integral, and plotted as a function of the load. A detailed stress analysis was performed and the critical stresses in the face sheet and the core were computed. The core was also molded as an isotropic elastic-perfectly plastic material and a nonlinear post buckling analysis was performed. A Graeco-Latin factorial plan was used to study the effects of debond length, fact sheet and core thicknesses, and core density on the load carrying capacity of the sandwich composite. It has been found that a linear buckling analysis is inadequate in determining the maximum load a debonded sandwich beam can carry. A nonlinear post-buckling analysis combined with an elasto-plastic model of the core is required to predict the compression behavior of debonded sandwich beams.

剥離を有する複合材料補強平板の圧縮挙動に関する数値解析

The present paper studies behaviors of blade-stiffened CFRP panels under axial compression by using a finite element method. Effect of a debonding between skin panel and stiffener flange on compressive behavior is analyzed and discussed. The debonding is supposed an impact damage. Linear buckling analysis and non-linear post-buckling analysis are conducted. At the debonded area, contact condition is approximately solved by an introduction of a spring element which has a resistant force only in the compressive direction. The some pre-buckling deflection increased proportional to the compression load before buckling owing to the asymmetric lamination at the flange portion, while the predicted linear buckling load agrees well with that obtained from the nonlinear analysis. The results well explain the experimental findings including the little reduction of compressive performance of the stiffened panel due to impact damage. The effect of the partial debonding on the compressive behavior becomes significant when the debonding area reaches the size comparable to that of the wave length of the buckling mode.

Postbuckling behavior of rectangular stiffened plates considering buckled pattern change

The postbuckling behavior of rectangular isotropic stiffened plates under inplane compressive load is investigated considering the phenomenon of buckled pattern change in the postbuckling load range. The finite deformation strains in the von Karman sense are incorporated in the geometrical nonlinear postbuckling analysis. The inplane and out-of-plane displacements are assumed as truncated Fourier series. The principle of minimum potential energy is applied. The total potential energy includes the strain energy due to bending and membrane action of plate, torsion of stiffeners, and the work done by external compressive force. The plates are assumed to be simply supported along the loaded sides and elastically restrained against rotation by longitudinal stiffeners along the unloaded sides. In numerical examples, the postbuckling load end-shortening curves and effective width curves are presented for the rectangular stiffened plates with various aspect ratios, torsional rigidities of stiffeners and inplane boundary conditions along unloaded sides.

Postbuckling performance of the TRIC natural mode triangular element for Isotropic and laminated composite shells

We present the computational performance and the achieved accuracy of the TRIC flat triangular shell element for nonlinear postbuckling analysis of arbitrary isotropic and composite shells. The element is based on the natural mode finite element method, which allows a convenient description of the current position of the structure. These natural modes are assigned to a convective coordinate system which follows the element during deformation within the framework of an Eulerian motion. With respect to this coordinate system the natural modes are additive. Numerical examples verify the accuracy, computational efficiency and the potential of the TRIC element in predicting the postbuckling behaviour of shells. Natural energy measures inform us about the energy allocation during nonlinear deformation and the interplay of the separate energy components.

Delamination buckling and postbuckling of composite cylindrical shells

A higher-order shear deformation theory is developed for accurately evaluating the transverse shear effects in delamination buckling and postbuckling of cylindrical shells under axial compression. The theory assures an accurate description of displacement field and the satisfaction of stress-free boundary conditions for the delamination problem. The governing differential equations of the present theory are obtained by applying the principle of virtual displacement. The Rayleigh-Ritz method is used to solve both linear and nonlinear equations by assuming a double trigonometric series for the displacements. Both linearized buckling analysis and nonlinear postbuckling analysis are performed for axially compressed cylindrical shells with clamped ends. Comparisons made with the classical laminate theory and a first-order theory show significant deviations.

Large deflection dynamic analysis of space frames

A robust numerical procedure based on the incremental secant stiffness concept for a large deflection static and dynamic analysis of space frames is described. The proposed method is capable of handling the nonlinear static post-buckling and dynamic analysis of space frames undergoing arbitrarily large rotations in three dimensional space efficiently and accurately. The technique can be easily implemented in an analysis program for framed structures to upgrade its capacity in predicting dynamic and nonlinear behaviour of frames. Numerical examples are given to illustrate the versatility of the computer method in handling the large deflection and dynamic behaviour of clamped beam, curved cantilever and space frame.

Nonlinear postbuckling analysis of plates and shells by four-noded strain element

This paper deals with the nonlinear postbuckling analysis of plates and shells by the finite element method. The four-noded C° strain element for nonlinear plate anaysis developed by the authors is first extended to the nonlinear shell analysis and then used for the nonlinear postbuckling analysis of plates and shallow shells. The element tangent stiffness matrix presented here is given explicitly, i.e., without any numerical integration. Consequently, this C° strain element is much more computationally economical and efficient than the widely used elements obtained from numerical integration. The efficiency and accuracy of the present four-noded strain element for postbuckling analysis are demonstrated by five numerical examples of various plates and shells.

Nonlinear and initial post-buckling finite element analysis of composite cylindrical panels

Nonlinear and initial post-buckling finite element analysis of composite cylindrical panels