On the crushing response of random open-cell foams

This study examines the effect of randomness of the cellular microstructure on the calculated compressive response of a class of open-cell aluminum alloy foams. The foams are modeled using realistic random soap froth with N³ cells generated using the Surface Evolver software. The ligaments are made straight but with non-uniform cross sectional area distributions that mimic those of the physical foams. The models are also assigned the density and anisotropy measured. The ligaments are modeled as shear deformable beams with the elasto-plastic material behavior of the Al-alloy. The microstructure is discretized with finite elements using LS-DYNA, which allows for beam-to-beam contact on the outer surface of the ligaments. 10³ cell domains compressed between rigid planes are shown to reproduce the measured compressive responses in both the rise and transverse directions. This includes the complete response from the initial elastic regime, through “yielding,” the extended stress plateau, to densification. More importantly, localized bands of crushed cells that develop and gradually spread throughout the domain resemble closely experimental observations made using X-ray tomography. This is a major improvement over previous models that idealized monodispersed foams as periodic Kelvin cells, and should allow modeling of polydisperse foams. The contact algorithm, friction between ligaments, and generally the discretization play crucial roles in the accuracy of the calculation as well as their numerical stability.     

On the compressive strength of open-cell metal foams with Kelvin and random cell structures

Two families of finite element models of anisotropic, aluminum alloy, open-cell foams are developed and their predictions of elastic properties and compressive strength are evaluated by direct comparison to experimental results. In the first family of models, the foams are idealized as anisotropic Kelvin cells loaded in the <100> direction and in the second family more realistic models, based on Surface Evolver simulations of random soap froth with N³ cells are constructed. In both cases the ligaments are straight but have nonuniform cross sectional area distributions that resemble those of the foams tested. The ligaments are modeled as shear deformable beams with elasto-plastic material behavior. The calculated compressive response starts with a linearly elastic regime. At higher stress levels, inelastic action causes a gradual reduction of the stiffness that eventually leads to a stress maximum, which represents the strength of the material. The periodicity of the Kelvin cell enables calculation of the compressive response up to the limit stress with just a single fully periodic characteristic cell. Beyond the limit stress, deformation localizes along the principal diagonals of the microstructure. Consequently beyond the limit stress the response is evaluated using finite size 3-D domains that allow the localization to develop. The random models consist of 3-D domains of 216, 512 or 1000 cells with periodicity conditions on the compressed ends but free on the sides. The compressive response is also characterized by a limit load instability but now the localization is disorganized resembling that observed in experiments. The foam elastic moduli and strengths obtained from both families of models are generally in very good agreement with the corresponding measurements. The random foam models yield 5–10% stiffer elastic moduli and slightly higher strengths than the Kelvin cell models. Necessary requirements for this high performance of the models are accurate representation of the material distribution in the ligaments and correct modeling of the nonlinear stress–strain response of the aluminum base material.     

Compressive response of open-cell foams. Part I: Morphology and elastic properties

This study is concerned with the understanding and modeling of the compressive response of open cell foams. The response starts with a nearly linear elastic regime which terminates into a limit load followed by an extensive load plateau. The plateau, which is responsible for the excellent energy absorption capacity of foams, is followed by a second stiff branch. Results from polyester urethane open cell foams with relative densities of about 0.025 are used to illustrate this behavior using experiments coupled with several levels of modeling. The experiments include characterization of the microstructure and the properties of the base material and measurement of the compressive response of the foams of various cell sizes.A sequence of models for predicting the complete response of such foam is developed. The foam is idealized to be periodic using the space-filling Kelvin cell assigned the major geometric characteristics found in the foams tested. The cells are elongated in the rise direction, the ligaments are assumed to be straight, to have Plateau border cross-sections and nonuniform cross-sectional area distribution. The ligaments are modeled as shear-deformable extensional beams and the base material is assumed to be linearly elastic. Prediction of the initial elastic moduli are addressed in Part I. Closed form expressions for the material constants are presented as well as results using a FE model of the characteristic cell. Comparison between measurements and predictions is very favorable. The paper finishes with results from a limited parametric study of the elastic moduli. The results demonstrate that inclusion of the geometric complexities mentioned above is essential for successful prediction of the moduli of such foams. The nonlinear parts of the response including the foam crushing behavior are addressed in Part II.     

On the microstructure of open-cell foams and its effect on elastic properties

Synthetic open-cell foams have a complex microstructure consisting of an interconnected network of cells resulting from the foaming process. The cells are irregular polyhedra with anywhere from 9 to 17 faces in nearly monodisperse foams. The material is concentrated in the nearly straight ligaments and in the nodes where they intersect. The mechanical properties of such foams are governed by their microstructure and by the properties of the base material. In this study micro-computed X-ray tomography is used to develop 3D images of the morphology of polyester urethane and Duocel aluminum foams with different average cell sizes. The images are used to establish statistically the cell size and ligament length distributions, material distributions along the ligaments, the geometry of the nodes and cell anisotropy. The measurements are then used to build finite element foam models of increasing complexity that are used to estimate the elastic moduli. In the most idealized model the microstructure is represented as a regular Kelvin cell. The most realistic models are based on Surface Evolver simulations of random soap froth with N³ cells in spatially periodic domains. In all models the cells are elongated in one direction, the ligaments are straight but have a nonuniform cross sectional area distribution and are modeled as shear deformable beams. With this input both the Kelvin cell models and the larger random foam models are shown to predict the elastic moduli with good accuracy but the random foams are 5–10% stiffer.     

ELASTO-PLASTIC DYNAMIC BUCKLING AND POSTBUCKLING OF SQUARE PLATES UNDER SOLID-FLUID SLAMMING

The elasto-plastic dynamic buckling and postbuckling phenomena ofsquare plates subjected to in-plane solid-fluid slamming are investigated.According tothe plate's response,the critical criteria for dynamic buckling,dynamic plasticity andplasti(?)collapse are defined,and the corresponding critical impulses are presented.Meanwhile,dynamic buckling modes and collapse models are observed.The effects ofdifferent boundary conditions and loading histories on the properties of buckling andpostbuckling are discussed.     

Secondary buckling of an elastic column with spring-supports at clamped ends

Summary The postbuckling behavior of an elastic column with spring supports of equal stiffness of extensional type at both clamped ends is studied. Attention is focused on those of spring stiffnesses near the critical value at which, under axial load, the column becomes critical with respect to two buckling modes simultaneously. By using the Liapunov-Schmidt-Koiter approach, we show that there are precisely two secondary bifurcation points on each primary postbuckling state for the spring stiffness greater than the critical value. The bifurcation takes place at one of the two least buckling loads. The corresponding secondary postbuckling states connect all the secondary bifurcation points in a loop. For the spring stiffness less than the critical value, no secondary bifurcation occurs. Asymptotic expansions of the primary and secondary postbuckling states are constructed. The stability analysis indicates that the primary postbuckling state for the spring stiffness greater than the critical value is bifurcating from the first buckling load and becomes unstable from a stable state via the secondary bifurcation, i.e., secondary buckling occurs. Received 22 April 1997; accepted for publication 22 December 1997     

Buckling and postbuckling of size-dependent cracked microbeams based on a modified couple stress theory

The elastic buckling analysis and the static postbuckling response of the Euler–Bernoulli microbeams containing an open edge crack are studied based on a modified couple stress theory. The cracked section is modeled by a massless elastic rotational spring. This model contains a material length scale parameter and can capture the size effect. The von Kármán nonlinearity is applied to display the postbuckling behavior. Analytical solutions of a critical buckling load and the postbuckling response are presented for simply supported cracked microbeams. This parametric study indicates the effects of the crack location, crack severity, and length scale parameter on the buckling and postbuckling behaviors of cracked microbeams.     

Postbuckling Behavior of Thin-Walled Open Cross-Section Compression Members

ABSTRACT

The postbuckling behavior of simply supported columns with thin-walled open cross section is investigated by means of the general nonlinear theory of elastic stability. Fourth-order terms in the series expansion of the total potential energy are disregarded.

It is shown that interaction between linearly independent simultaneous buckling modes is responsible for neutral equilibrium at bifurcation if the column cross section has two axes of symmetry, and unstable if it has only one. If the buckling modes are not simultaneous, the equilibrium is neutral in both cases. Finally, the equilibrium at bifurcation is usually unstable if the cross section has no axis of symmetry.     

Non-linear in-plane postbuckling of arches with rotational end restraints under uniform radial loading

This paper presents a thorough and comprehensive investigation of non-linear buckling and postbuckling analyses of pin-ended shallow circular arches subjected to a uniform radial load and which have equal elastic rotational end-restraints. The differential equations of equilibrium for non-linear buckling and postbuckling are established based on a virtual work approach. Exact solutions for the non-linear bifurcation, limit point and lowest buckling loads are obtained; in particular, exact solutions for the non-linear postbuckling equilibrium paths are derived. The criteria for switching between fundamental buckling and postbuckling modes are developed in terms of critical values of a geometric parameter for an arch, with exact solutions for these critical values of geometric parameter being obtained. Analytical solutions of non-linear buckling and postbuckling problems for arches with rotational end-restraints are of great interest, since they constitute one of the very few closed-form analyses of buckling and postbuckling behaviour of continuous structural systems. These exact solutions are a contribution to the non-linear structural mechanics of arches, as well as providing useful benchmark solutions for verifying non-linear numerical analyses.     

Dynamic crushing of aluminum foams: Part II – Analysis

Part II of this study uses micromechanically accurate foam models to simulate and study the dynamic crushing of open-cell foams. The model starts as random soap froth generated using the Surface Evolver software to mimic the microstructure of the foams tested. The linear edges of the cellular microstructure are “dressed” with appropriate distributions of solid to match those of ligaments in the actual foams and their relative density. The ligaments are modeled as shear-deformable beams with variable cross sections discretized with beam elements in LS-DYNA, while the Al-alloy is modeled as a finitely deforming elastic–plastic material. The numerical contact algorithm of the code is used to model ligament contact and limit localized cell crushing. The quasi-static and all dynamic crushing experiments in Part I are simulated numerically. The models are shown to reproduce all aspects of the crushing behavior including the formation and evolution of nearly planar shocks, the force acting at the two ends, the shock front velocity, the strain in the crushed material behind the shock, and the energy absorbed.     

Energy method solution for the postbuckling response of an axially loaded bilaterally constrained beam

Bistable and multistable structures have shown great usefulness in many applications such as MEMS actuation and energy harvesting. Bistability of structures can be achieved through buckling. Confining a buckled beam between two lateral constraints allows it to buckle into higher modes as the axial load increases. This paper presents a theoretical study of the postbuckling response of a bilaterally constrained elastica subjected to gradually increased axial load. Equilibrium states are determined using an energy method. Under small deformation assumptions, the total potential energy is minimized under the defined constraints. The presented model allows for an accurate representation of the flatting behavior and the increase in the length of contact areas with the lateral constraints before the sudden snapping between equilibrium states. Mode transitions are manifested by jumps in the response curves. Previously developed models based on geometry and symmetries overestimate the required forces for higher equilibrium modes and do not match experimental observations. Results are validated with experimental force–displacement measurements under both force- and displacement-control. The kinetic energy released during buckling mode transitions is determined by a dynamic analysis.     

On the crushing of aluminum open-cell foams: Part II analysis

Part II of this study is concerned with the modeling of all aspects of the compressive response and crushing of the open-cell Al foam studied in Part I. The foam microstructure is modeled using the regular cell of Kelvin with cell anisotropy and ligament geometry established by X-ray tomography. The ligaments are modeled as shear-deformable beams and the material is elastoplastic calibrated to the properties of the Al alloy base material. It is demonstrated that the initiation stress of measured responses is associated with a limit load instability that results from plastification of foam ligaments due to combined bending and axial compression. The periodicity of the Kelvin cell enables calculation of the initial elastic properties as well as the initiation stress with just a single fully periodic characteristic cell. The crushing response is evaluated by considering finite size 3D domains that allow localized deformation to develop. Localization is in the form of shear buckling that develops along the principal diagonals of the Kelvin cell foam. Localized crushing is arrested by contact between the ligaments of the buckled cells. Contact is approximated by limiting the amount a cell can collapse in the direction of the applied load. This arrests local collapse and causes it to spread to neighboring material at a nearly constant stress level as in the experiments. The stress picks up when the whole domain has crushed. Although the calculated collapse patterns differed from the more random ones observed in the experiments, the calculated force–displacement responses match very well the experimental ones in all aspects.     

The Effect of Multiple Buckling Modes on the Postbuckling Behavior of Plane Elastic Frames. Part I. Symmetric Frames

The postbuckling behavior of a one-bay, two-storey frame with built-in edges and symmetric with respect to midspan is analyzed. Columns are assumed to be inextensible and shear-undeformable, and beams are rigid. Then two buckling modes are possible, that is, sidesway of the lower floor with rigid horizontal displacement of the top floor and sidesway of the top floor with the lower floor undergoing no displacement. Obviously, the two buckling modes occur simultaneously if the ratios EI/h⁷ (EI being the bending stiffness of a column and h its length) are properly selected. Within the framework of a Koiter-type energy approach a suitable perturbation formulation is derived from a “hybrid” functional which is obtained by adding to the potential energy certain extra terms which account for the nonlinear energy associated with the internal forces applied to the beam at the joints. Results show that the postbuckling behavior of a single buckling mode can be stable or unstable according to the value of the ratio h/l, where l is the frame span. In the case of simultaneous buckling modes the structural behavior in the postbuckling range never improves, but no severe changes are noticed in comparison with the preceding case.     

Postbuckling analysis of elastic shells of revolution considering mode switching and interaction

The postbuckling response of shells is known to exhibit complex phenomena including mode switching and interaction, particularly in the advanced postbuckling range. The existing literature contains many initial postbuckling analyses as well as advanced postbuckling analyses for a single buckling mode, but little work is available on the advanced postbuckling analysis of shells of revolution considering mode switching and interaction. In this paper, a numerical method for the advanced postbuckling analysis of thin shells of revolution subject to torsionless axisymmetric loads is presented, in which such mode switching and interaction are properly captured. Numerical results obtained using the present method for several typical problems not only demonstrate the capability of the method, but also lead to significant observations concerning the postbuckling behavior of thin shells of revolution. In particular, the results show that strong interaction between different harmonic modes may exist and the transition of deformation mode from one to another is gradual. Consequently, the conventional approach of finding the postbuckling path of a shell as the lower festoon curve of postbuckling paths of individual harmonic modes is not valid and is at best a convenient approximation.     

Exact solution and stability of postbuckling configurations of beams

We present an exact solution for the postbuckling configurations of beams with fixed–fixed, fixed–hinged, and hinged–hinged boundary conditions. We take into account the geometric nonlinearity arising from midplane stretching, and as a result, the governing equation exhibits a cubic nonlinearity. We solve the nonlinear buckling problem and obtain a closed-form solution for the postbuckling configurations in terms of the applied axial load. The critical buckling loads and their associated mode shapes, which are the only outcome of solving the linear buckling problem, are obtained as a byproduct. We investigate the dynamic stability of the obtained postbuckling configurations and find out that the first buckled shape is a stable equilibrium position for all boundary conditions. However, we find out that buckled configurations beyond the first buckling mode are unstable equilibrium positions. We present the natural frequencies of the lowest vibration modes around each of the first three buckled configurations. The results show that many internal resonances might be activated among the vibration modes around the same as well as different buckled configurations. We present preliminary results of the dynamic response of a fixed–fixed beam in the case of a one-to-one internal resonance between the first vibration mode around the first buckled configuration and the first vibration mode around the second buckled configuration.     

Buckling and postbuckling of a compressed thin film bonded on a soft elastic layer: a three-dimensional analysis

The wrinkling of a stiff thin film bonded on a soft elastic layer and subjected to an applied or residual compressive stress is investigated in the present paper. A three-dimensional theoretical model is presented to predict the buckling and postbuckling behavior of the film. We obtained the analytical solutions for the critical buckling condition and the postbuckling morphology of the film. The effects of the thicknesses and elastic properties of the film and the soft layer on the characteristic wrinkling wavelength are examined. It is found that the critical wrinkling condition of the thin film is sensitive to the compressibility and thickness of the soft layer, and its wrinkling amplitude depends on the magnitude of the applied or residual in-plane stress. The bonding condition between the soft layer and the rigid substrate has a considerable influence on the buckling of the thin film, and the relative sliding at the interface tends to destabilize the system.     

Postbuckling and imperfection sensitivity of fixed-end and free-end struts on elastic foundation

Summary A study of the postbuckling and imperfection sensitivity of fixed-end and free-end struts on a Winkler elastic foundation is carried out. The configuration and stability of the postbuckling paths bifurcating from the critical points are analysed. For the most part of foundation stiffness, the corresponding postbuckling paths are shown to be falling with respect to load and be unstable. This indicates that, for almost all values of foundation stiffness, the buckling loads of the struts will be sensitive to imperfections. We also obtain imperfection sensitivity of the struts with respect to geometric imperfections having the shape of buckling modes. Received 30 October 1998, accepted for publication 30 March 1999     

Postbuckling analysis and its application to stretchable electronics

A versatile strategy for fabricating stretchable electronics involves controlled buckling of bridge structures in circuits that are configured into open, mesh layouts (i.e. islands connected by bridges) and bonded to elastomeric substrates. Quantitative analytical mechanics treatments of the responses of these bridges can be challenging, due to the range and diversity of possible motions. Koiter (1945) pointed out that the postbuckling analysis needs to account for all terms up to the 4th power of displacements in the potential energy. Existing postbuckling analyses, however, are accurate only to the 2nd power of displacements in the potential energy since they assume a linear displacement–curvature relation. Here, a systematic method is established for accurate postbuckling analysis of beams. This framework enables straightforward study of the complex buckling modes under arbitrary loading, such as lateral buckling of the island-bridge, mesh structure subject to shear (or twist) or diagonal stretching observed in experiments. Simple, analytical expressions are obtained for the critical load at the onset of buckling, and for the maximum bending, torsion (shear) and principal strains in the structure during postbuckling.     

Thermal buckling and postbuckling of laminated composite beams with temperature-dependent properties

The thermal buckling and postbuckling analysis of laminated composite beams with temperature-dependent material properties is presented. The governing equations are based on the first-order shear deformation beam theory (FSDT) and the geometrical nonlinearity is modeled using Green's strain tensor in conjunction with the von Karman assumptions. The differential quadrature method (DQM) as an accurate, simple and computationally efficient numerical tool is adopted to discretize the governing equations and the related boundary conditions. A direct iterative method is employed to obtain the critical temperature (bifurcation point) as well as the nonlinear equilibrium path (the postbuckling behavior) of symmetrically laminated beams. The applicability, rapid rate of convergence and high accuracy of the method are established via different examples and by comparing the results with those of existing in literature. Then, the effects of temperature dependence of the material properties, boundary conditions, length-to-thickness ratios, number of layers and ply angle on the thermal buckling and postbuckling characteristic of symmetrically laminated beams are investigated.     

Koiter circles in the buckling of axially compressed conical shells

As is well known, the elastic stability of shell structures under certain loading conditions is characterised by a severely unstable postbuckling behaviour. The presence of simultaneous buckling modes (‘competing’ modes corresponding to the same critical buckling load) is deemed to be largely responsible for such a behaviour. In the present paper, within the framework of the so-called classical theory (linear bifurcation eigenvalue analysis), the buckling behaviour of axially compressed cylindrical shells is firstly reviewed. Accordingly, doubly periodic eigenvectors (buckling modes) corresponding to the same eigenvalue (critical buckling load) can be determined, and their locus in a dimensionless meridional and circumferential buckling wavenumber space is described by a circle (known as the Koiter circle). In the case of axially compressed conical shells, no clear evidence of the existence of simultaneous buckling modes can be found in the literature. Then, such a problem is studied here via linear eigenvalue finite element analyses, showing that simultaneous doubly periodic modes do also occur for cones, and that their locus in a specifically defined dimensionless wavenumber space can be described by an ellipse (hereafter termed as the Koiter ellipse) whose aspect ratio is dependent on the tapering angle of the cone.