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.     

Simulation of the densification of real open-celled foam microstructures

Ubiquitous in nature and finding applications in engineering systems, cellular solids are an increasingly important class of materials. Foams are an important subclass of cellular solids with applications as packing materials and energy absorbers due to their unique properties. A better understanding of foam mechanical properties and their dependence on microstructural details would facilitate manufacture of tailored materials and development of constitutive models for their bulk response. Numerical simulation of these materials, while offering great promise toward furthering understanding, has also served to convincingly demonstrate the inherent complexity and associated modeling challenges.The large range of deformations which foams are subjected to in routine engineering applications is a fundamental source of complication in modeling the details of foam deformation on the scale of foam struts. It requires accurate handling of large material deformations and complex contact mechanics, both well established numerical challenges. A further complication is the replication of complex foam microstructure geometry in numerical simulations. Here various advantages of certain particle methods, in particular their compatibility with the determination of three-dimensional geometry via X-ray microtomography, are exploited to simulate the compression of “real” foam microstructures into densification. With attention paid to representative volume element size, predictions are made regarding bulk response, dynamic effects, and deformed microstructural character, for real polymeric, open-cell foams. These predictions include a negative Poisson's ratio in the stress plateau, and increased difficulty in removing residual porosity during densification.     

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.     

通孔泡沫铝的动态压缩行为

在SHPB装置上对渗流法制备的通孔泡沫铝进行了动态压缩实验,研究了相对密度为0.341~0.419的通孔泡沫铝在10-3~2000 s-1应变率范围内的压缩响应特征和应变率相关性,并用扫描电镜（scanning electron microscope,SEM)分析了泡沫铝的压缩变形特征。实验结果表明,通孔泡沫铝有明显应变率效应,随应变率上升,泡沫铝流动应力提高。SEM观察结果揭示,在动态压缩下,通孔泡沫铝宏观上均匀变形,微观变形机制以泡孔横向伸展坍塌为主。     

Deformation rate effects on failure modes of open-cell Al foams and textile cellular materials

The compressive behavior of open-cell aluminum alloy foam and stainless steel woven textile core materials have been investigated at three different deformation rate regimes. Quasi-static compressive tests were performed using a miniature loading frame, intermediate rates were achieved using a stored energy Kolsky bar, and high strain rate tests were performed using a light gas gun.In agreement with previous studies on foam materials, the strain rate was not found to have a significant effect on the plateau stress of metallic foams. For all the tests, real time imaging of the specimen combined with digital image correlation analysis allowed the determination of local deformation fields and failure modes. For the Kolsky bar tests, the deformations in the foam specimen were found to be more distributed than for the quasi-static test, which is attributed to moderate inertia effects. The differences in failure mode are more dramatic for the gas gun experiments, where a full compaction shock wave is generated at the impact surface. The stresses in front and behind the shock wave front were determined by means of direct and reverse gas gun impact tests, i.e., stationary and launched specimen, respectively. A one-dimensional shock wave model based on an idealized foam behavior is employed to gain insight into the stress history measurements. We show that the predictions of the model are consistent with the experimental observations. Woven textile materials exhibited moderate dependence of strength on the deformation rate in comparison with open-cell foam materials.     

Permeability of open-cell foams under compressive strain

A model for the behavior of low-density, open-cell foam under compressive strain is proposed. Using this model, a tractable relationship between the normalized permeability and the applied strain is developed. An experimental study of the effect of strain on the permeability of open-cell polyurethane foams is presented. The experiments are performed using a Newtonian fluid in the fully laminar regime, where viscous forces are assumed to dominate. The model is found to describe the experimental data well and be independent of the foam cell size, the direction of flow with respect to the foam rise direction, and the properties of the saturating fluid. In a companion paper, the model for the permeability of open-cell foam is combined with Darcy’s law to give the contribution of viscous fluid flow to the stress–strain response of a reticulated foam under dynamic loading.     

Effects of microstructure on uniaxial strength asymmetry of open-cell foams

Based on the elongated Kelvin model, the effect of microstructure on the uniaxial strength asymmetry of open-cell foams is investigated. The results indicate that this asymmetry depends on the relative density, the solid material, the cell morphology, and the strut geometry of open-cell foams. Even though the solid material has the same tensile and compressive strength, the tensile and compressive strength of open-cell foams with asymmetrical sectional struts are still different. In addition, with the increasing degree of anisotropy, the uniaxial strength as well as the strength asymmetry increases in the rise direction but reduces in the transverse direction. Moreover, the plastic collapse ratio between two directions is verified to depend mainly on the cell morphology. The predicted results are compared with Gibson and Ashby''s theoretical results as well as the experimental data reported in the literature, which validates that the elongated Kelvin model is accurate in explaining the strength asymmetry presented in realistic open-cell foams.     

Mechanical characterization of the role of defects in sintered FeCrAlY foams

Open celled metal foams fabricated through metal sintering are a new class of material that offers novel mechanical and acoustic properties. Previously, polymer foams have been widely used as a means of absorbing acoustic energy. However, the structural applications of these foams are limited. The metal sintering approach offers a cost- effective means for the mass-production of open-cell foams from a range of materials, including high-temperature steel alloys. In this first part of two-paper series, the mechanical properties of open-celled steel alloy (FeCrAlY) foams were characterized under uniaxial compression and shear loading. Compared to predictions from established models, a significant knockdown in material properties was observed. This knockdown was attributed to the presence of defects throughout the microstructure that result from the unique fabrication process. Further in situ tests were carried out in a SEM (scanning electronic microscope) in order to investigate the effects of defects on the properties of the foams. Typically, the onset of plastic yielding was observed to occur at defect locations within the microstructure. At lower relative densities, ligament bending dominates, with the deformation initializing at defects. At higher relative densities, an additional deformation mechanism associated with membrane elements was observed. In the follow-up of this paper, a finite element model will be constructed to quantify the effects of defects on the mechanical performance of the open-cell foam. The project supported by the US Office of Naval Research (N000140210117), the National Basic Research Program of China (2006CB601202), the National Natural Science Foundation of China (10328203, 10572111, 10632060), and the National 111 Project of China (B06024).     

Effective elastic behavior of some models for perfect cellular solids

The effective elastic behavior of some models for low density cellular solids, or solid foams, are calculated using analytical and numerical techniques. The models are perfect in the sense that imperfections or irregularities as often encountered in real foams have been removed. We believe that the present models can serve as references to which more advanced models which include imperfections and irregularities can be compared. The work in this paper does not address buckling or yielding in cell walls, which play an increasingly important role as foam stresses increase.     

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.     

Indentation into polymeric foams

The effects of the nose shape of rigid indenters on the indentation behaviour of polymethacrylimide (PMI) and polyetherimide (PEI) foams with different densities are investigated. Experimental results show that indentation resistance depends on the geometry of the indenter and the density of the foam. Analytical models based on the deformation mechanisms observed in experiments are developed to predict the indentation resistance. It shows that the analytical predictions are in good agreement with experimental measurements for a range of polymeric foams. This study presents a complete and systematic experimental data on the indentation behaviours of a range of polymeric foams and demonstrates the capability of the analytical model to predict the indentation behaviours of PMI and PEI foams.     

Kelvin结构开孔泡沫材料的弹性性能研究

采用一修正的十四面体结构模型(Kelvin结构模型)对开孔泡沫金属的弹性性能进行研究,对低密度开孔泡沫材料表现出不可压的特性进行了分析。该模型考虑作用在泡沫筋条上的弯矩、剪力和轴向力,以及轴向力的平衡。修正模型的数值计算结果与实验结果及其他模型的结果进行了对比,结果表明修正模型计算的杨氏模量比原有模型的略有提高,筋条截面为星形的修正模型计算的结果与实验比较符合。在密度等同的条件下,筋条截面惯性矩越大的开孔泡沫材料,其弹性模量也越大,而泊松比则越小。Kelvin结构的开孔泡沫材料的泊松比随相对密度的减小而趋于0.5。     

The interaction between shock waves and foam in a shock tube

An experimental study and a numerical simulation were conducted to investigate the mechanical and thermodynamic processes involved in the interaction between shock waves and low density foam. The experiment was done in a stainless shock tube (80 mm in inner diameter, 10 mm in wall thickness and 5 360 mm in length). The velocities of the incident and reflected compression waves in the foam were measured by using piezo-ceramic pressure sensors. The end-wall peak pressure behind the reflected wave in the foam was measured by using a crystal piezoelectric sensor. It is suggested that the high end-wall pressure may be caused by a rapid contact between the foam and the end-wall surface. Both open-cell and closed-cell foams with different length and density were tested. Through comparing the numerical and experimental end-wall pressure, the permeability coefficients α and β are quantitatively determined.     

Analytical models for the mechanical behavior of closed and open-cell Al foams

Two 3D analytical models are proposed for the determination of mechanical properties of Al closed and open-cell foams under compression load. The first model, referring to closed cell foams, is symmetrical, considering ellipsoid cells equally arranged in a rectangular plate, whereas the second one, related to open-cell foams, consists of a simple unit parallelepiped cell.The closed cell model produces much higher values of the plateau stress than comparable experimental results, mainly due to the associated conditions of symmetry, contrary to the open-cell model which yields values close to experimental and theoretical results of other investigators. Additionally, in the latter case, a unit cubic cell is also considered for comparison reasons. Both models are solved by the finite element method using a commercial program. The open-cell model is simple, time sparing and easy to use. Finally, a fracture analysis of the model is conducted based on the energy density concept and results are given for distortion and dilatational effects.     

Experimental measurements and numerical modeling of dynamic compression response of an interpenetrating phase composite foam

A 3D interpenetrating phase composite (IPC) foam made by infusing syntactic foam (SF) into an open-cell aluminum scaffold is studied under high-strain rate conditions. The SF is prepared by dispersing hollow glass microballoons into an epoxy matrix. Dynamic compression characteristics, including strength and failure progression of pure SF and IPC foam made of four different volume fractions of microballoons in SF measured using a split Hopkinson pressure bar (SHPB) apparatus are reported. The results show that in general IPC foams outperform SF in terms of compressive strength. An idealized 3D elasto-plastic unit-cell based finite element (FE) model is proposed for studying the IPC foam. The 3D geometry of the aluminum constituent in the cubic unit-cell is modeled as a tetrakaidecahedron, whereas the SF constituent is modeled as the occupant of the rest of the unit-cell. The computational model incorporates infinite elements to represent the far-field regions surrounding the unit-cell. Comparisons of computational and experimental results for different microballoon volume fractions of SF are reported.     

Modeling of the role of defects in sintered FeCrAlY foams

The metal sintering approach offers a cost- effective means for the mass-production of open-cell foams from a range of materials, including high-temperature steel alloys, which offer novel mechanical and acoustic properties. In a separate experimental study, the mechanical properties of open-celled steel alloy (FeCrAlY) foams have been characterized under uniaxial compression and shear loading. Compared to predictions from established models, a significant knockdown in material properties was observed. This knockdown was attributed to the presence of defects throughout the microstructure that result from the unique fabrication process. In the present paper, the microstructure of sintered FeCrAlY foams was modeled by using a finite element (FE) model. In particular, microstructural variations were introduced to a base lattice, and the effects on the strength and stiffness calculated. A range of defects identified under scanning electronic microscope (SEM) imaging were considered including broken ligaments, thickness variations, and pore blockages, which are the three primary imperfections observed in sintered foams. The corresponding levels of defect present in the material were subsequently input into the FE model, with the resulting predictions correlating well with experimental data. The project supported by the National Basic Research Program of China (2006CB601202), the National Natural Science Foundation of China (10328203, 10572111, 10632060), the National 111 Project of China (B06024), and the US Office of Naval Research (N000140210117).     

Predicting pressure drop in open-cell foams by adopting Forchheimer number

Interconnected struts arranged in 3-D foam structures pose a challenge in understanding fluid flow, which is significantly different from that in traditional porous media. Different flow regimes (Darcy, transition and weak inertia regimes) and thus, different flow laws in open-cell foams are used. The impact of characteristic lengths’ choices based on both, morphological and hydraulic parameters on flow law formulation has been studied. Ambiguities in definitions and measurements of several key parameters have been shown and limitations in the use of some parameters have been pointed out.An equivalent Reynolds number in the form of Forchheimer number (F_o) has been proposed to establish the friction factor relationship in order to avoid any morphological ambiguities. This number takes into account hydraulic characteristics of viscous and inertia regimes simultaneously. It has been observed that when F_o < 0.1, the flow through open-cell foams remains in the Darcy regime while the occurrence of weak inertia regime dominates when F_o > 1. Transition regime occurs in a narrow range of flow velocity when 0.1 < F_o < 1. The limits of transition for regime identification are found to be independent of foam morphologies. The form drag coefficient varies in relation with foam morphological parameters and is not a “universal” constant.Empirical correlations have been derived to predict hydraulic characteristics and friction factor data for different strut shapes and porosities. An excellent agreement has been obtained between predicted and numerical/experimental flow data.     

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.     

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。     

Impact of material processing and deformation on cell morphology and mechanical behavior of polyurethane and nickel foams

The cell morphology and mechanical behavior of open-cell polyurethane and nickel foams are investigated by means of combined 3D X-ray micro-tomography and large scale finite element simulations. Our quantitative 3D image analysis and finite element simulations demonstrate that the strongly anisotropic tensile behavior of nickel foams is due to the cell anisotropy induced by the deformation of PU precursor during the electroplating and heat treatment stages of nickel foam processing. In situ tensile tests on PU foams reveal that the initial main elongation axis of the cells evolves from the foam sheet normal direction to the rolling direction of the coils. Finite element simulations of the hyperelastic behavior of PU foams based on real cell morphology confirm the observation that cell struts do not experience significant elongation after 0.15 tensile straining, thus pointing out alternative deformation mechanisms like complex strut junctions deformation. The plastic behavior and the anisotropy of nickel foams are then satisfactorily retrieved from finite element simulations on a volume element containing eight cells with a detailed mesh of all the hollow struts and junctions. The experimental and computational strategy is considered as a first step toward optimization of process parameters to tailor anisotropy of cell shape and mechanical behavior for applications in batteries or Diesel particulate filtering.