A SIMPLISTIC ANALYTICAL MODEL OF PERMEABILITY FOR OPEN-CELL METALLIC FOAMS

针对通孔金属泡沫中的渗透率预测及现有理论模型的局限性,发展了一种新的全解析渗透率模型. 该模型以立方体结构作为代表单元,采用基于追踪流体微团轨迹的分支算法解析求解代表单元内的流动迂曲度. 渗透率的表达形式简单且不含任何拟合或经验参数,仅是孔隙率与平均孔径的函数. 采用实验测量和文献数据对模型预测进行了验证. 结果表明:提出的模型能够在较为宽广的孔隙率(0.55~0.98) 和孔密度(5~100 PPI) 范围内预测孔通孔金属泡沫的渗透率;采用分支算法得到的流动迂曲度能够较好地描述流体在通孔金属泡沫中的流动特征;采用开孔率修正的解析模型亦能对半开孔泡沫材料的渗透率提供良好预测.     

通孔金属泡沫渗透率解析模型

针对通孔金属泡沫中的渗透率预测及现有理论模型的局限性,发展了一种新的全解析渗透率模型. 该模型以立方体结构作为代表单元,采用基于追踪流体微团轨迹的分支算法解析求解代表单元内的流动迂曲度. 渗透率的表达形式简单且不含任何拟合或经验参数,仅是孔隙率与平均孔径的函数. 采用实验测量和文献数据对模型预测进行了验证. 结果表明:提出的模型能够在较为宽广的孔隙率(0.55~0.98) 和孔密度(5~100 PPI) 范围内预测孔通孔金属泡沫的渗透率;采用分支算法得到的流动迂曲度能够较好地描述流体在通孔金属泡沫中的流动特征;采用开孔率修正的解析模型亦能对半开孔泡沫材料的渗透率提供良好预测.      

Experimental study on the two-phase pressure drop in copper foams

Experiments were performed to study pressure drops in copper foams embedded in a rectangular copper channel. De-ionized water was used as the working fluid with mass fluxes of 30–200 kg/m² s, and inlet temperature of 40–80°C. The copper foam has the porosity of 0.88 and the pore densities of 30, 60 and 90 ppi (pores per inch). Both single-phase liquid flow and boiling two-phase flow are studied. Effects of mass fluxes, vapor mass qualities, and average pore diameters of metallic foams are investigated. It is found that friction factors for the single-phase liquid flow are mainly dependent on the Reynolds number and the average pore diameter of metallic foams. The friction factors are decreased with increases in the Reynolds numbers, and will approach 0.22 at high Reynolds numbers. For the boiling two-phase flow, two-phase pressure drops are increased with increases in the outlet vapor mass qualities, mass fluxes, and ppi values. The two-phase multiplier is increased with increases in the outlet vapor mass qualities and mass fluxes, and it is decreased with increases in the Martinelli parameter and will attain a constant value depending on the mass fluxes. The larger the mass fluxes, the larger the constant value is. An experimental correlation considering the effects of vapor mass qualities, mass fluxes, and average pore diameters of metallic foams is recommended, showing good accuracy to predict the two-phase pressure drops in metallic foams.     

多孔介质中稳定泡沫的流动计算及实验研究

将多孔介质简化为一簇变截面毛管束,根据多孔介质的颗粒直径、颗粒排列方式、孔喉尺度比及束缚水饱和度,计算出变截面毛细管的喉道半径和孔隙半径. 在考虑多孔介质喉道和孔隙中单个气泡的受力和变形基础上,利用动量守恒定理,推导出单个孔隙单元内液相的压力分布和孔隙单元两端的压差计算公式,最终得到多孔介质的压力分布计算公式. 利用长U型填砂管对稳定泡沫的流动特性进行了实验研究. 研究结果表明:稳定泡沫流动时多孔介质中的压力分布呈线性下降,影响泡沫在多孔介质中流动特性的因素包括:多孔介质的孔喉结构、泡沫流体的流量和干度、气液界面张力、气泡尺寸,其中孔喉结构和泡沫干度是影响泡沫封堵能力的主要因素.关键词: 稳定泡沫;多孔介质;变截面毛管;流动;表观粘度;压力分布;实验研究      

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).     

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.     

Effects of internal pore pressure on closed-cell elastomeric foams

A micromechanics framework for porous elastomers with internal pore pressure (Idiart and Lopez-Pamies, 2012) is used together with an earlier homogenization estimate for elastomers containing vacuous pores (Lopez-Pamies and Ponte Castañeda, 2007a) to investigate the mechanical response and stability of closed-cell foams. Motivated by applications of technological interest, the focus is on isotropic foams made up of a random isotropic distribution of pores embedded in an isotropic matrix material, wherein the initial internal pore pressure is identical to the external pressure exerted by the environment (e.g. atmospheric pressure). It is found that the presence of internal pore pressure significantly stiffens and stabilizes the response of elastomeric foams, and hence that it must be taken into account when modeling this type of materials.     

Modeling of metallic foam morphology using the Representative Volume Element approach: Development and experimental validation

This paper focuses on the development of an algorithm capable of generating morphologically-representative foam structures using the Representative Volume Element (RVE) approach. Stereology, a sampling method based on direct observations of the foam cross-sections, is used to characterize the pore size and shape distributions. Using the morphology generation algorithm, the smallest RVEs corresponding to the numerically-convergent foam morphologies are calculated for different foam porosities. To validate the foam generation algorithm, the pore size and shape distributions of the numerically-generated foams are compared to those of the titanium foams manufactured by the space holder method.     

Measurements of pore scale flows within and exiting ceramic foams

LDA was used to measure the mean velocities and pore scale turbulence intensity of air flowing through ceramic foams having pore sizes between 0.8 and 2.5 mm in diameter. The stream-wise and transverse velocity components were measured near the exit plane, in a region between two foams separated by 5 mm, and within a section of ceramic foam. Mean velocities ranged from 0.25 m/s to 1.5 m/s, corresponding to pore Reynolds numbers between 20 and 200. A transition to a fully developed pore turbulence was achieved at pore Reynolds numbers above approximately 150.Support for this work was provided by the Texas Advanced Research Program and The Center for Energy Studies of The University of Texas at Austin.     

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.     

低密度开孔泡沫蠕变松弛特性分析

利用三维Voronoi模型和有限元方法分析了胞壁材料具有粘弹特性的低密度开孔泡沫的蠕变和应力松弛行为.采用了三参量标准线性固体模型来描述胞壁材料的粘弹特性.所得结果表明.低密度开孔泡沫具有与其胞壁材料相同的松弛时间,当相对密度较低时(低于1%)开孔泡沫的松弛模量与胞壁材料的松弛模量和泡沫相对密度平方成正比.此外,计算结果还表明,低密度开孔泡沫在较小的初始应力条件下具有与其胞壁材料相同的延迟时间.其蠕变柔度与胞壁材料的蠕变柔度和泡沫相对密度平方倒数基本成正比.但随着初始应力值的增大,泡沫的延迟时间将会显著增加.     

DISCRETE MODELLING OF TWO-DIMENSIONAL LIQUID FOAMS

Liquid foam is a dense random packing of gas or liquid bubbles in a small amount of immiscible liquid containing surfactants. The liquid within the Plateau borders, although small in volume, causes considerable difficulties to the investigation of the spatial structure and physical properties of foams, and the situation becomes even more complicated as the fluid flows. To solve these problems, a discrete model of two-dimensional liquid foams on the bubble scale is proposed in this work. The bubble surface is represented with finite number of nodes, and the liquid within Plateau borders is discretized into lattice particles. The gas in bubbles is treated as ideal gas at constant temperatures. This model is tested by choosing an arbitrary shape bubble as the initial condition. This then automatically evolves into a circular shape, which indicates that the surface energy minimum routine is obeyed without calling external controlling conditions. Without inserting liquid particle among the bubble channels, periodic ordered and disordered dry foams are both simulated, and the fine foam structures are developed. Wet foams are also simulated by inserting fluid among bubble channels. The calculated coordination number, as a function of liquid fractions, agrees well with the standard values.     

低密度开孔泡沫材料力学模型的理论研究进展

开孔泡沫材料主要用于隔音、减振和填充方面，对其力学行为进行理论描述，探讨力学性能与密度及复杂微结构的关系具有十分重要的学术价值和工程意义．为了促进国内泡沫材料力学的发展和交流，文中对低密度开孔泡沫材料力学模型的研究历史进行了简要回顾，重点介绍了能较好地反映开孔泡沫材料真实胞体结构特点的十四面体胞体模型和随机胞体模型，并报道了近年来基于十四面体胞体模型和随机胞体模型研究低密度开孔泡沫材料力学行为的一些理论工作、同时，也对国内的一些相关研究情况进行了简要评述，指出了该领域今后的一些研究方向．     

泡沫铝材料准静态本构关系的理论和实验研究

应用Chen和Lu提出的适用于可压缩弹塑性固体的唯象本构模型框架，建立了泡沫铝的准静态本构模型，推导了三维等比例加载和环向受约束轴向加载下的宏观应力-应变曲线. 对两种泡沫铝材料(开孔和闭孔)进行了4类准静态试验，即单轴压缩、三维静水压缩、三维等比例压缩和侧向受约束轴向压缩实验. 利用单轴压缩和三维静水压缩的实验结果得到了泡沫铝材料的本构参数曲线，并由此预测它在三维等比例压缩和侧向受约束轴向压缩情况下的响应. 理论预测与相应实验结果相比较，三维等比例压缩的结果比较吻合，但与侧向受约束轴向压缩的结果却相差很大. 分析表明，理论预测与侧向受约束轴向压缩实验结果的偏差是由于泡沫铝试件与约束筒之间的摩擦造成的. 研究结果说明, Chen-Lu模型能够很好地描述泡沫铝材料在压缩占主导的应力状态下的响应.     

STOCHASTIC ELASTO-PLASTIC FRACTURE ANALYSIS OF ALUMINUM FOAMS

Model I quasi-static nonlinear fracture of aluminum foams is analyzed by considering the effect of microscopic heterogeneity. Firstly, a continuum constitutive model is adopted to account for the plastic compressibility of the metallic foams. The yield strain modeled by a two- parameter Weibull-type function is adopted in the constitutive model. Then, a modified cohesive zone model is established to characterize the fracture behavior of aluminum foams with a cohesive zone ahead of the initial crack. The tensile traction versus local crack opening displacement relation is employed to describe the softening characteristics of the material. And a Weibull statistical model for peak bridging stress within the fracture process zone is used for considering microscopic heterogeneity of aluminum foams. Lastly, the influence of stochastic parameters on the curve of stress-strain is given. Numerical examples are given to illustrate the numerical model presented in this paper and the effects of Weibull parameters and material properties on J-integral are discussed.     

Effective Thermal Conductivity Modelling for Closed-Cell Porous Media with Analytical Shape Factors

This paper presents a fully analytical model for the effective thermal conductivity of two-phase porous media with two-/three-dimensional closed cells, applicable to honeycombs and closed-cell foams. The present model combines an existing analytical expression derived based on the Laplace heat conduction equation with an analytical shape factor which corrects the deviation caused from a non-circular (or non-spherical) pore inclusion. Results demonstrate the validity of the present model capable of analytically estimating the effective thermal conductivity of closed-cell porous media. The simple yet accurate model provides the physical mechanisms of how effective thermal conductivity depends upon the shape of pores.     

Micro-structural Impact of Different Strut Shapes and Porosity on Hydraulic Properties of Kelvin-Like Metal Foams

Various ideal periodic isotropic structures of foams (tetrakaidecahedron) with constant ligament cross section are studied. Different strut shapes namely circular, square, diamond, hexagon, star, and their various orientations are modeled using CAD. We performed direct numerical simulations at pore scale, solving Navier–Stokes equation in the fluid space to obtain various flow properties namely permeability and inertia coefficient for all shapes in the porosity range, \(0.60<\varepsilon <0.95\) for wide range of Reynolds numbers, \(10^{-6} . We proposed an analytical model to obtain pressure drop in metallic foams in order to correlate the resulting macroscopic pressure and velocity gradients with the Ergun-like approach. The analytical results are fully compared with the available numerical data, and an excellent agreement is observed.     

Hydrodynamic Characterization of Nickel Metal Foam,Part 2: Effects of Pore Structure and Permeability

In this article, the structural characterization of chemical vapor deposition (CVD) nickel metal foam is presented. Scanning electron microscope and post image processing were used to carefully analyze the surface of the nickel metal foams. Data on foam unit cell, ligament thickness, projected pore diameter, and averaged porosity was obtained. Unit cell and projected pore diameters of CVD nickel metal foam possess Gaussian-like distribution. Characteristics of pore structure and its effect on permeability in Darcian flow regime were analyzed. The relations between the permeability, pore size, and porosity are presented. Present and previous data are compared with these relations. Measurement results indicate that the permeability or the viscous conductivity of the CVD processed metal foam is affected not only by the pore size, and porosity but also by the ligament structure.     

A micromechanical model for predicting the fracture toughness of functionally graded foams

In this paper, finite element method based micromechanical analysis is used to understand the fracture behavior of functionally graded foams. The finite element analysis uses a micro-mechanical model in conjunction with a macro-mechanical model in order to relate the stress intensity factor to the stresses in the struts of the foam. The stress intensity factor at the crack tip of the macro-mechanical model can be evaluated using either the J-contour integral or the stresses in the singularity-dominated zone. The fracture toughness is evaluated for various crack positions and length within the functionally graded foam. Then the relationship between the fracture toughness of the graded foam and the local density at the crack tip is studied. Convergence tests for both macro-mechanical and micro-mechanical model analysis were conducted in order to maintain adequate accuracy with reasonable computational time. Fracture toughness of homogenous foams and functionally graded foams for various cases are presented as a function of relative density. This study indicates that the fracture toughness of functionally graded foams mainly depends on the relative density at the crack-tip.