The effect of strut geometry on the yielding behaviour of open-cell foams

A micromechanical analysis was carried out to investigate the effect of strut geometry on the yielding behaviour of open-cell foams. Different strut cross sections, in rectangular, circular and equilateral triangular shapes, were investigated. It was found that the strut geometry significantly affects the plastic-yielding behaviour of open-cell foams. The shape of the plastic-yield surface was found to depend not only on relative density but also on the cross-sectional shape of the struts. Numerical results show that even though the material of the struts is perfectly plastic, open-cell foams with asymmetrical sectional struts will exhibit different tensile and compressive collapse strength.     

Uniaxial strength asymmetry in cellular materials: An analytical model

The ability to predict failure of cellular materials depends on the knowledge of microstructural mechanisms that contribute to macroscopic behavior. In this paper, we develop microstructural models to examine the mechanisms responsible for differences in tensile and compressive strength observed in cellular materials. We limit our analyses to those materials that fail by the same mechanism (yielding or microcracking) in tension and compression. Using both a honeycomb and an open-cell foam model, we demonstrate that density-dependent, compression-strong strength asymmetry arises when two conditions are met: the cell wall material has a higher yield strength in compression than in tension, and the cell walls are loaded simultaneously by axial forces and bending moments. Our models predict that strength asymmetry (defined as the ratio of compressive to tensile yield strength of the cellular material) increases with relative density (as observed in real materials such as rigid polyurethane foams and trabecular bone), and that strength asymmetry is more pronounced in anisotropic materials (where oblique struts are more closely aligned with the direction of loading).     

Effective mechanical behavior of hyperelastic honeycombs and two-dimensional model foams at finite strain

The aim of the present study is the analytical and numerical determination of the effective stress–strain behavior of solid foams made from hyperelastic materials in the finite strain regime. For the homogenization of the microstructure, a strain energy-based concept is proposed which assumes macroscopic mechanical equivalence of a representative volume element for the given microstructure with a similar homogeneous volume element if the strain energy of both volume elements is equivalent, provided that the volume average of the deformation gradient is equal for both volume elements. The concept is applied to an analysis of hyperelastic solid foams using a two-dimensional model. The effective stress–strain behavior is analyzed under uniaxial and biaxial loading conditions in the tensile and in the compressive range as well as under simple shear deformation. It is observed that the effective mechanical behavior of cellular solids at infinitesimal and finite deformation is essentially different on both, the quantitative and the qualitative level.     

Stiffness of bi-modulus hexagonal and diamond honeycombs

Cellular materials are widely used in various applications because of their low density and high strength. The mechanical behavior of cellular materials under various loading conditions has been investigated. Nevertheless, many of these previous studies assume that the Young’s modulus of constituting struts is the same in tension and compression. The present work first derives analytical expressions for the effective Young’s moduli of hexagonal and diamond lattices composed of struts with different tension and compression moduli under the assumption of small strain deformation. It also uses the finite element method to further investigate the mechanical responses of these lattices. The macroscopic Young’s moduli under both compressive and tensile loads are reported as a function of the ratio of compression and tension moduli of constituting struts. The numerical finite element models are implemented by a user defined material subroutine in ABAQUS. Results reveal that the effective Young’s moduli of periodic hexagonal and diamond lattices significantly decrease with decreasing ratio of compression and tension moduli of the struts. Furthermore, the mechanical behavior of hexagonal lattices composed of struts with different tension-compression moduli is dependent on the loading direction and whether they are compressed or stretched. The unique mechanical properties of bi-modulus cellular materials could find important applications in the automotive and construction industries.

     

A general constitutive relation for linear elastic foams

A three-dimensional linear elastic constitutive relation is formulated based on a representative unit cell of foam using elasticity theory and micromechanics homogenization scheme. The displacement and strain fields of the unit cell are obtained from elasticity theory and used to derive the macroscopic strain field defined on the outer surface of unit cell through homogenization scheme. By assuming a uniform macroscopic stress on the unit cell surface and the existence of strain energy potential, the constitutive relation of linear elastic foams is obtained. The newly derived constitutive relation is a function of mechanical property of solid constituent, the geometry of cell struts, and the porosity of foams and is able to characterize the anisotropic behavior of foams due to non-uniform strut geometry. The linear elastic response of open-celled foams with both low- and medium-relative densities can be studied using the derived constitutive relation. The effective elastic modulus for uniform strut geometry is reduced from the constitutive relation and agrees well with Gibson and Ashby's semi-empirical equation, Warren and Kraynik's, and Zhu's analytical models within relative density ranging from 0 to 0.35. For non-uniform strut geometry, the calculated effective elastic moduli in three axial directions are different and the material displays anisotropic behavior. The bulk modulus shows less dependence on the variation of the strut geometry. Poisson's ratios are also reduced from the compliance matrix.     

泡沫铝结构对其拉伸力学性能的影响

研究了开孔与闭孔两种胞孔结构的泡沫铝在不同相对密度下的准静态拉伸力学性能,并与单向压缩性能进行了对比.结果表明：开孔和闭孔泡沫铝的拉伸曲线由线弹性变形段和塑性变形段组成,线弹性变形段很短,塑性屈服中没有出现明显的屈服点;高密度的开孔泡沫铝的杨氏模量、抗拉强度较低密度的闭孔泡沫铝要大;随着相对密度的增大,两种结构泡沫铝的力学性能均明显增强,符合Gibson和Ashby关系式,泡沫铝在准静态下的抗拉强度比抗压强度略低,而拉伸杨氏模量比压缩杨氏模量大得多.     

基于十四面体模型的开孔泡沫材料弹性模量的有限元分析

利用十四面体模型描述开孔泡沫材料的胞体结构,并用有限元方法确定开孔泡沫材料的弹性模量。计算中使用相同尺寸的十四面体胞体模型,并考虑两种不同支柱截面（圆截面和Plateau截面）形状对弹性模量计算的影响。此外,通过数值方法研究开孔泡沫材料的弹性模量随模型尺寸的变化规律。同时,讨论边界条件处理对开孔泡沫材料弹性模量计算的影响。结果表明,支柱为Plateau截面形状的模型,其弹性模量明显高十具有圆截面支柱模型的结果。且两种模型的弹性模量均随模型尺度的增加而增加,最终趋于一个稳定值,并与Warren和Kraynik的理论预测较为一致。此外,边界条件对模型刚度的影响随着模型尺度的增加而逐渐减小。     

A constitutive model for polyurethane foam with strain rate sensitivity

The present work investigates the strain rate dependent behavior of polyurethane foams and formulates a new constitutive model in order to improve the fit of the experimental data at various strain rates. The model has seven parameters that are decided by quasi-static compression tests at two strain rates. Two models for low and high density polyurethane foams are shown to give stress strain relation at various strain rates. Dynamic compression tests were carried out to give stress strain data at high strain rate and the results are compared with those of the constitutive model.     

Long-wave buckling of elastic square honeycombs subject to in-plane biaxial compression

In this study, long-wave and short-wave buckling of elastic square honeycombs subject to in-plane biaxial compression are analyzed using a two-scale theory of the updated Lagrangian type. By taking cell aggregates to be periodic units, the bifurcation and post-bifurcation behavior are analyzed so that the dependence of buckling stress on periodic length can be discussed. It is shown that buckling stress decreases as periodic length increases, and that very-long-wave buckling occurs just after the onset of macroscopic instability if the periodic length is sufficiently long. Then, a simple formula to evaluate the very-long-wave buckling stress under in-plane biaxial compression is derived by exploring the macroscopic instability condition in the light of the two-scale analysis. The resulting formula is verified using an energy method.     

微结构几何形状对开孔泡沫金属屈服行为的影响

利用理想弹塑性梁在拉弯载荷作用下的屈服条件研究筋条截面形状为等边三角形和圆形的开孔泡沫金属在两轴载荷作用下的的屈服轨迹 ,并与现有数学模型的结果进行对比。理论分析结果表明筋条截面形状对泡沫金属的屈服行为有很大影响 ,特别是对筋条截面为非对称结构的泡沫金属 ;泡沫金属的屈服轨迹形状不但依赖于材料的相对密度 ,而且还与筋条截面形状密切相关。     

A new hardening rule for metallic foam plasticity

Metallic foams are a class of porous materials widely used in the industry because of their advantages. In recent years, extensive studies on the behavior of these materials have been conducted. Several constitutive equations have also been presented and applied. This study proposes a new constitutive equation that predicts metallic foam behavior using the stress–strain curve in uniaxial compression. The proposed model offers a new functionality for work hardening and is evaluated for both isotropic and combined hardening. The constitutive equations are implemented in MATLAB and integrated using return mapping algorithm. The material parameters are identified using genetic algorithm and through a comparison of the experimental and numerical results. The aluminum foams discussed in this paper are the commercially available types, Foaminal and Alporas. The comparison of numerical and experimental results indicate that this new constitutive equation predicts foam behavior in a reasonable manner. Moreover, a good agreement is observed between the experimental and computational curves.     

Characterization of micro- to macroscopic deformation behavior of amorphous polymer with heterogeneous distribution of microstructures

In the present study, we clarify the micro- to macroscopic deformation behavior of an amorphous polymer with a slightly heterogeneous distribution of molecular chains; in other words, the distribution of the initial shear strength of the polymer. The micro- to macroscopic deformation behaviors of polymer under macroscopically uniform tension and shearing, uniaxial extension of a plane strain block and surface deformation of the plane strain block under compression were investigated by means of computational simulation with the nonaffine molecular chain network model. The results revealed the onset of microscopic shear bands emanating from slightly weak points and their evolution, and the interaction and percolation of new shear bands. The effects of distribution patterns and standard deviation of initial shear strength on the deformation, the interaction of weak points, the transition from microscopic shear band formation to macroscopic neck propagation and the evolution of surface undulation under compression have been demonstrated.     

Discussion of cyclic plasticity and viscoplasticity of single crystal nickel-based superalloy in large strain analysis: comparison of anisotropic macroscopic model and crystallographic model

The inelastic behavior of nickel-based superalloy is investigated in detail by application of a macroscopic anisotropic plasticity model developed here, and the results are compared to predictions based on crystal plasticity, which incorporates the kinematic hardening. Uniaxial deformation processes and simple shear deformations at large strain are considered. The plastic spin concept coupled with an anisotropic Chaboche model is provided in the framework of macroscopic viscoplasticity. The plastic spin formulation used here is based on the concept of the noncoaxiality between the stress and plastic rate of deformation. The present model succeeds in reproducing the inelastic behavior during large deformation. It is shown that the plastic spin associated with the anisotropic flow rule plays a key role in the macroscopic model. Simulation results find these two different scale models provide similar predictions under uniaxial deformation for [0 0 1] and [1 1 1] orientation, while their predictions for simple shear deformation at large strain exhibit quantitative difference, but their trends are the same. The interpretations for simulation results are pursued in detail.     

Reproducing hoop stress–strain behavior for tubular material using lateral compression test

Identification of material properties in the hoop direction, such as stress–strain behavior, is essential in tube hydroforming processes. Conventional tests such as uniaxial tension and compression tests have some drawbacks and limitations. In the current investigations a simple technique to identify the stress–strain behavior in the hoop direction for tubular material is introduced, based on the experimental data obtained from tube lateral compression test. In the proposed technique, an assumed stress–strain curve is used in finite element simulation to predict the load deflection curve of the tube lateral compression. An iterative algorithm is used to compare the calculated and experimental load deflection curves until a good agreement with a percentage deviation less than 4% is obtained. The suggested technique was used to obtain the material properties of Cu–40%Zn brass tube. The predicted stress–strain curve was compared with that obtained from uniaxial compression test. Comparison between the experimental and predicted stress–strain curve showed that the proposed technique is effective in the prediction of the material properties from the tube lateral compression test with percentage deviation less than 1%.     

开孔弹性泡沫材料拉伸变形过程的数值模拟

为了研究低密度弹性开孔泡沫材料的拉伸力学性能,文中建立具有不同胞体几何特性的泡沫材料随机模型,并用有限元方法模拟该材料的拉伸变形过程.同时,从材料的微结构层次上分析弹性开孔泡沫材料拉伸变形的机制,以及相对密度和胞体几何性质对拉伸非线性力学行为的影响.并且,将拉伸应力-应变曲线与Warren和Kraynik的四面体微结构模型的预测结果进行对比,说明数值模拟较好地反映开孔泡沫材料的拉伸变形特征.     

杆系结构鲁棒性应变能敏感度分析及试验

为研究结构鲁棒性动态分析方法,进行了理论分析、数值模拟和破坏性模型试验。结合平面悬臂桁架和六角星型穹顶算例,利用应变能敏感度法分析了杆系结构鲁棒性。基于杆件移除法讨论了结构应变能和应变能敏感度对结构破坏特征的影响,并同鲁棒性应力变化率评价法进行了对比。结果表明:应变能敏感度法能动态地反映杆系结构鲁棒性变化、预测结构稳定极限荷载、识别结构破坏模态及评价杆件易损性;尽管支座压杆、环向拉杆对应变能敏感度的影响比中心压杆小,但可能引起结构整体倾覆和中心压杆强度破坏;工程应用中应在杆件重要性分析和结构破坏模态识别基础上采用应变能敏感度法评价结构鲁棒性。     

闭孔泡沫铝力学特性及其在汽车碰撞吸能中的应用研究进展

汽车低能耗、安全和轻量化已经成为汽车领域研究的热点问题,闭孔泡沫铝作为一种轻质吸能金属材料,在低密度下具有良好的比刚度和比强度,同时具有良好的抗冲击性和能量吸收性,已逐渐引起汽车产业界地重视。简述泡沫铝单轴压缩试验中弹性模量、抗压强度、屈服强度、平台应力、致密化应变等参数的定义和试验标准;综述闭孔泡沫铝的本构方程的研究现状,重点讨论屈服面模型;总结泡沫铝的微观结构有限元建模方法,比较商业软件中集成的宏观材料模型。归纳吸能材料的特点,分析闭孔泡沫铝的吸能能力和抗冲击能力;综述应变率和冲击速度对泡沫铝吸能特性有无影响的研究进展,并对可能存在的影响进行解释。总结闭孔泡沫铝在汽车轻量化和碰撞安全性领域的应用,具体分析典型的案例。指出当前闭孔泡沫铝的力学特性及其在汽车结构中应用存在的问题与难点,总结并提出本研究领域可以借鉴的研究方向。     

Measurement of strain rate sensitivity of aluminium foams for energy dissipation

In this paper the structural performance of aluminium alloy foams have been investigated under both static and dynamic compression loads. Three foam typologies (M-PORE, CYMAT, SCHUNK) in a wide range of density (from 0.14 to 0.75 g/cm³), made by means of different process-routes (melt gas injection, powder metallurgy, investment casting) have been analysed. Foams microstructural characterization has been carried out through morphometric measurements by means of Scanning Electron Microscopy (SEM) and Computed Tomography (CT) and subsequent digital image processing in order to determine average cells size and cell distributions on different section planes. The experimental study aims to assess the strain rate sensitivity and energy absorption capability of commercially available metal foams and to point out the correlation between the mechanical behaviour and the physical and geometrical properties of the foam. It has been found that the specific energy dissipation of foams with similar density can be quite different: for the same volume of foam, average values of 1770, 1780 and 5590 J/kg at 50% nominal compression have been measured on M-PORE (0.19 g/cm³), CYMAT (0.28 g/cm³) and SCHUNK (0.28 g/cm³) foams, respectively. Impact tests showed that the dependence of the plateau stress on strain rate could be considered negligible for M-PORE and CYMAT foams while it is quite remarkable for SCHUNK foams. Moreover, it was found that the peak stress of CYMAT foams has a quite large sensitivity on the loading rate.     

Application of a continuum constitutive model to metallic foam DEN-specimens in compression

The behavior of double-edge notched specimens of metallic foams in compression is studied numerically. To model the constitutive behavior of the metallic foam, a recently developed phenomenological, pressure-sensitive yield surface [1] is used. Compressive yielding in response to hydrostatic stress is incorporated through a dependence on the plastic Poisson ratio ν^p. Results are presented in terms of limit load F_lim, as a function of notch depth, a/W, and the plastic Poisson ratio ν^p. For incompressible plastic behavior, ν^p=0.5, the results show notch-strengthening due to constrained plastic deformation near the crack/notch-tip. For fully compressible plastic behavior (no lateral expansion on uniaxial compression, ν^p=0), no notch-effect is observed. The validity of using a continuum model for the analysis of metallic foam notched specimens is discussed.