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.     

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.     

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.     

On the crushing of aluminum open-cell foams: Part I. Experiments

This two-part study is concerned with the understanding and modeling of the compressive response of open-cell metallic foams. Part I presents experimental results from Al-6101-T6 foams of three different cell sizes with relative densities of about 8%. X-ray tomography is first used to characterize the geometry of the microstructure. The cells are irregular polyhedra of nearly uniform size that are somewhat elongated in one direction. The ligaments are nearly straight with convex, three-sided cross-sections and variable area distribution along their length. Foam specimens were compressed at slow displacement rates along the rise and transverse directions and the evolution of crushing in the specimens was monitored using X-ray tomography. In both directions, the response is initially nearly linear, terminating into a limit load that is followed by an extensive load plateau. At an average strain of about 55% the load increases monotonically again due to densification. The limit load is caused by plastification due to combined compression and bending of the ligaments. Beyond this point, cells start to buckle and collapse locally, forming bands that cover the full cross-section of the specimen. Contact of the collapsing cells arrests local deformation triggering collapse in neighboring cells. In this manner, crushing gradually spreads throughout the specimen and when this is achieved the load required for further deformation starts to rise. The initial elastic modulus, the stresses at the limit load and the plateau and the extent of the plateau have been measured as a function of relative density for both directions. The stress–displacement response in the transverse direction is generally somewhat lower than in the rise direction but the prevalent events were found to be similar in the two directions.     

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.     

Three-dimensional modeling of the mechanical property of linearly elastic open cell foams

Three-dimensional Voronoi models are developed to investigate the mechanical behavior of linearly elastic open cell foams. Dependence of the Young’s modulus, Poisson’s ratio and bulk modulus of the foams on the relative density is evaluated through finite element analysis. Obtained results show that in the low density regime the Young’s modulus and bulk modulus of random Voronoi foams can be well represented by those of Kelvin foams, and are sensitive to the geometric imperfections inherent in the microstructure of foams. In contrast, the compressive plateau stress of the foams is less sensitive to the imperfections. Failure surface of the foams subject to multi-axial compression is determined and is found to comply with the maximum compressive principal stress criterion, consistent with available experimental observations on polymer foams. Numerical results also show that elastic buckling of cell edges at microscopic level is the dominant mechanism responsible for the compressive failure of elastic open cell foams.     

Modeling and prediction of bulk properties of open-cell carbon foam

The emerging ultralightweight material, carbon foam, was modeled with three-dimensional microstructures to develop a basic understanding in correlating microstructural configuration with bulk performance of open-cell foam materials. Because of the randomness and complexity of the microstructure of the carbon foam, representative cell ligaments were first characterized in detail at the microstructural level. The salient microstructural characteristics (or properties) were then correlated with the bulk properties through the present model. In order to implement the varying anisotropic nature of material properties in the foam ligaments, we made an attempt to use a finite element method to implement such variation along the ligaments as well as at a nodal point where the ligaments meet. The model was expected to provide a basis for establishing a process-property relationship and optimizing foam properties.The present model yielded a fairly reasonable prediction of the effective bulk properties of the foams. We observed that the effective elastic properties of the foams were dominated by the bending mode associated with shear deformation. The effective Young's modulus of the foam was strongly influenced by the ligament moduli, but was not influenced by the ligament Poisson's ratio. The effective Poisson's ratio of the foam was practically independent of the ligament Young's modulus, but dependent on the ligament Poisson's ratio. The effective Young's modulus of the carbon foam was dependent more on the transverse Young's modulus and the shear moduli of the foam ligaments, but less significantly on the ligament longitudinal Young's modulus. A parametric study indicated that the effective Young's modulus was significantly improved by increasing the solid modulus in the middle of the foam ligaments, but nearly invariant with that at the nodal point where the ligaments meet. Therefore, appropriate processing schemes toward improving the transverse and shear properties of the foam ligaments in the middle section of the ligaments rather than at the nodal points are highly desirable for enhancing the bulk moduli of the carbon foam.     

On the stability of Kelvin cell foams under compressive loads

It has been previously shown that the nonlinearity exhibited in the compressive response of open cell foams is governed by cell ligament buckling. Significant insight into this behavior can be gained by idealizing such foams as periodic, space-filling Kelvin cells assigned several of the geometric characteristics of actual foams. 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 mechanical response of such foams can be established using models of a characteristic cell assigned appropriate periodicity conditions. The ligaments are modeled as shear deformable beams. The periodicity of this microstructure allows the use of Bloch wave theory to conduct the search for the critical state efficiently. The method tailored to the present microstructure is outlined. It is subsequently used to establish the critical states for uniaxial and a set of triaxial loadings. A rich variety of buckling modes are identified which are affected by the anisotropy and the mutliaxiality of the applied loads. Under some loadings the critical modes have long wavelengths which are shown to lead to unstable postbuckling behavior involving localization. Under other loading conditions the modes are either local to the characteristic cell or involve an assemblage of a few such cells. For the cases analyzed local modes were found to have a stable postbuckling response.     

基于三维细观力学模型的开孔泡沫弹性性能预测

泡沫材料的宏观力学性能主要取决于基体材料的力学特性及其微细观结构特征,基于细观力学模型的分析方法是泡沫材料力学性能研究的重要途径。文中基于Matlab语言和Abaqus软件构建了描述中等孔隙率开孔弹性泡沫材料微结构特征的三维随机分布球形泡孔模型,并采用有限元方法对弹性泡沫压缩变形进行了模拟,并计算给出了不同孔隙率弹性泡沫材料弹性模量、剪切模量、体积模量以及泊松比的分布,建立了相应的唯象表达式。与理论模型及测试结果的比较表明,本文基于三维随机泡孔模型模拟结果构建的唯象表达式能够对弹性泡沫材料的弹性力学性能给出很好的预测。     

The elastic–plastic behaviour of foam under shock loading

An experimental investigation of the elastic–plastic nature of shock wave propagation in foams was undertaken. The study involved experimental blast wave and shock tube loading of three foams, two polyurethane open-cell foams and a low-density polyethylene closed-cell foam. Evidence of precursor waves was observed in all three foam samples under various compressive wave loadings. Experiments with an impermeable membrane are used to determine if the precursor wave in an open-cell foam is a result of gas filtration or an elastic response of the foam. The differences between quasi-static and shock compression of foams is discussed in terms of their compressive strain histories and the implications for the energy absorption capacity of foam in both loading scenarios. Through a comparison of shock tube and blast wave loading techniques, suggestions are made concerning the accurate measurements of the principal shock Hugoniot in foams.     

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.     

The high strain mechanical response of the wet Kelvin model for open-cell foams

Surface Evolver software was used to create the three-dimensional geometry of a Kelvin open-cell foam, to simulate that of polyurethane flexible foams. Finite Element Analysis (FEA) with 3D elements was used to model large compressive deformation in the [0 0 1] and [1 1 1] directions, using cyclic boundary conditions when necessary, treating the polyurethane as an elastic or elastic–plastic material. The predicted foam Young’s moduli in the [0 0 1] direction are double those of foams with uniform Plateau border cross-section edges, for the same foam density and material properties. For compression in the [1 1 1] direction, the normalized Young’s modulus increases from 0.9 to 1.1 with foam relative density, and the predicted stress–strain relationship can have a plateau, even for a linearly-elastic polymer. As the foam density increases, the predicted effects of material plasticity become larger. For foam of relative density 0.028, edge-to-edge contact is predicted to occur at a 66% strain for [1 1 1] direction compression. The foam is predicted to contract laterally when the [1 1 1] direction compressive strain exceeds 25%.     

Modeling the dynamic response of visco-elastic open-cell foams

This article introduces a mesoscopic formulation for modeling the dynamic response of visco-elastic, open-cell solid foams. The effective material response is obtained by enforcing on a representative 3D unit cell the principle of minimum action for dissipative systems. The resulting model accounts explicitly for the foam topology, the elastic and viscous properties of the cell wall, and the inertial effects arising from non-affine motion within the cells. The microinertial effects become significant in retarding the foam collapse during exceedingly high strain-rate loading. As an application example, a heterogenous case of compressive deformation at high strain rate is simulated utilizing the present model as a constitutive update in a non-linear finite element analysis code. This FEM simulation shows the ability of the model to capture the progressive foam collapse during the dynamic compression as observed in experimental studies. Using the microscopic model, the inertial and viscous strain-rate effects are investigated through the foam density, viscosity, and relative density. Based on the physics incorporated into the local cell model, we provide insight into the physical mechanisms responsible for the experimentally observed strain-rate effects on the behavior of dynamically loaded foam materials.     

Modelling open cell-foams based on the Weaire–Phelan unit cell with a minimal surface energy approach

The complex architecture of open cell foams has most often been described by Kelvin cell models. It has been shown that the accuracy to predict the elastic properties of open cell foams increases with an increasing level of detail and resemblance to real foam microstructures. However, the Kelvin cell does not possess pentagonal faces which are the most abundant within real open cell foams. Therefore this study focuses on the use of the Weaire–Phelan unit cell to model the elastic properties of an open cell polyurethane foam. Optical and scanning electron microscopy were used to characterise the architecture of the open cell foam. Surface Evolver software was used to minimize the surface energy and introduce the typical architectural characteristics of the open cell foam to the FE-model. The E-modulus and Poisson coefficient of the Kelvin and Weaire–Phelan cell show a similar behaviour as a function of density. The Weaire–Phelan cell predicts however a higher dependency of the shear modulus on the density. When the influence of the elongation of the cells in the rise direction of the foam and the uncertainty of the solid material properties of the polyurethane is taken into account, a good accuracy of the Kelvin cell and Weaire–Phelan structure based FE-models versus experimental compression tests is found.     

Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams

The Voronoi tessellation technique and the finite element (FE) method are utilized to investigate the microstructure-property relations of three-dimensional (3-D) cellular solids (foams) that have irregular cell shapes and non-uniform strut cross-sectional areas (SCSAs). Perturbations are introduced to a regular packing of seeds to generate a spatially periodic Voronoi diagram with different degrees of cell shape irregularity (amplitude a), and to the constant SCSA to generate a uniform distribution of SCSAs with different degrees of SCSA non-uniformity (amplitude b). Twenty FE models are constructed, based on the Voronoi diagrams for twenty foam samples (specimens) having the same pair of a and b, to obtain the mean values and standard deviations of the elastic properties. Spatially periodic boundary conditions are applied to each specimen. The simulation results indicate that for low-density imperfect foams, the elastic moduli increase as cell shapes become more irregular, but decrease as SCSAs get less uniform. When the relative density (R) increases, the elastic moduli of imperfect foams increase substantially, while the Poisson's ratios decrease moderately. The effect of the interaction between the two types of imperfections on foam elastic properties appears to be weak. In addition, it is found that the strut cross-sectional shape has a significant effect on the foam properties. Also, the elastic response of foams with the cell shape and SCSA imperfections appears to be isotropic regardless of changes in a, b and R and the strut cross-sectional shape.     

铝/硅橡胶复合材料动态压缩行为的研究

通过向开孔泡沫铝中填充硅橡胶而制备了铝 /硅橡胶复合材料 ,在Hopkinson压杆实验装置上对这种材料进行了动态压缩实验 ,分析了其动态压缩应力 应变响应特征 ,并与开孔结构泡沫铝的压缩行为进行了比较。结果表明 :铝 /硅橡胶复合材料的压缩应力 应变响应具有两个阶段的特征 ,即弹性和塑性变形阶段 ;这种复合材料具有较强的应变率效应 ,随应变率的提高 ,其屈服强度和流动应力显著上升。     

Dynamic compressive strength properties of aluminium foams. Part I—experimental data and observations

This study of the dynamic compressive strength properties of metal foams is in two parts. Part I presents data from an extensive experimental study of closed-cell Hydro/Cymat aluminium foam, which elucidates a number of key issues and phenomena. Part II focuses on modelling issues.The dynamic compressive response of the foam was investigated using a direct-impact technique for a range of velocities from 10 to . Elastic wave dispersion and attenuation in the pressure bar was corrected using a deconvolution technique.A new method of locating the point of densification in the nominal stress-strain curves of the foam is proposed, which provides a consistent framework for the definition of the plateau stress and the densification strain, both essential parameters of the ‘shock’ model in Part II. Data for the uniaxial, plastic collapse and plateau stresses are presented for two different average cell sizes of approximately 4 and 14 mm. They show that the plastic collapse strength of the foam changes significantly with compression rate. This phenomenon is discussed, and the distinctive roles of microinertia and ‘shock’ formation are described. The effects of compression rates on the initiation, development and distribution of cell crushing are also examined. Tests were carried out to examine the effects of density gradient and specimen gauge length at different rates of compression and the results are discussed. The origin of the conflicting conclusions in the literature on the correlation between nominal strain rate (ratio of the impact velocity V_i to the initial gauge length l_o of the specimen) and the dynamic strength of aluminium alloy foams is identified and explained.     

Multiscale mass-spring models of carbon nanotube foams

This article is concerned with the mechanical properties of dense, vertically aligned CNT foams subject to one-dimensional compressive loading. We develop a discrete model directly inspired by the micromechanical response reported experimentally for CNT foams, where infinitesimal portions of the tubes are represented by collections of uniform bi-stable springs. Under cyclic loading, the given model predicts an initial elastic deformation, a non-homogeneous buckling regime, and a densification response, accompanied by a hysteretic unloading path. We compute the dynamic dissipation of such a model through an analytic approach. The continuum limit of the microscopic spring chain defines a mesoscopic dissipative element (micro-meso transition) which represents a finite portion of the foam thickness. An upper-scale model formed by a chain of non-uniform mesoscopic springs is employed to describe the entire CNT foam. A numerical approximation illustrates the main features of the proposed multiscale approach. Available experimental results on the compressive response of CNT foams are fitted with excellent agreement.     

双模量面板泡沫铝芯圆形层合板的非线性弯曲

双模量材料是典型的拉压弹性模量不同的材料,在均匀外载荷作用下,双模量面板泡沫铝芯圆形层合板相当于三种不同材料组成的层合板。采用弹性理论建立了双模量面板泡沫铝芯圆形层合板在均布载荷作用下的静力平衡方程,利用该静力平衡方程确定了层合板的中性面位置。在此基础上建立了双模量面板泡沫铝芯圆形层合板的大挠度弯曲微分方程组,求得了层合板中心挠度与均布载荷的关系式。该方法计算结果与有限元计算结果的最大误差仅为3.8%,这说明该方法是可靠的。算例分析表明不考虑面板拉压弹性模量相异时其计算结果与实际情况相差较大,超过了工程上所允许的计算误差5%。所以,在计算双模量面板泡沫铝芯圆形层合板的非线性弯曲时,不宜采用相同弹性模量弹性理论,而应该采用拉压弹性模量不同的弹性理论。     

填充硅橡胶的泡沫铝复合材料的力学性能

用渗流法向开孔泡沫铝-硅合金和泡沫纯铝中充填硅橡胶获得含硅橡胶的泡沫材料, 在材料试验机和SHPB上对含硅橡胶的复合材料进行动态与准静态压缩实验。实验结果表明:含硅橡胶的泡沫复合材料只有弹性段和塑性段两个阶段,具有更高的应变率敏感性,其应力-应变曲线抖动幅度比较大。