Stress analysis of two-dimensional cellular materials with thick cell struts

Finite element analyses (FEA) were performed to thoroughly validate the collapse criteria of cellular materials presented in our previous companion paper. The maximum stress (von-Mises stress) on the cell strut surface and the plastic collapse stress were computed for two-dimensional (2D) cellular materials with thick cell struts. The results from the FEA were compared with those from theoretical criteria of authors. The FEA results were in good agreement with the theoretical results. The results indicate that when bending moment, axial and shear forces are considered, the maximum stress on the strut surface gives significantly different values in the tensile and compressive parts of the cell wall as well as in the two loading directions. Therefore, for the initial yielding of ductile cellular materials and the fracture of brittle cellular materials, in which the maximum stress on the strut surface is evaluated, it is necessary to consider not only the bending moment but also axial and shear forces. In addition, this study shows that for regular cellular materials with the identical strut geometry for all struts, the initial yielding and the plastic collapse under a biaxial state of stress occur not only in the inclined cell struts but also in the vertical struts. These FEA results support the theoretical conclusion of our previous companion paper that the anisotropic 2D cellular material has a truncated yield surface not only on the compressive quadrant but also on the tensile quadrant.     

Plastic collapse of cellular structures comprised of doubly tapered struts

In this paper, micro-structural models are developed to examine the effects of tapered strut morphology on the plastic collapse of cellular structures. The analytical models are for materials that fail by plastic yielding and cover several types of columnar structure (e.g. hexagonal and square honeycombs, and hexagonal and rhombic cellular materials of rod-like columnar structure). The results indicate that the plastic collapse of hexagonal cellular materials is dependent not only on the relative density but also on the strut morphology. The presence of taper in struts can increase or decrease the plastic collapse strength of cellular materials, depending on the strut morphology.     

A morphological elastic model of general hexagonal columnar structures

A general three-dimensional (3D) anisotropic hexagonal model of columnar structure with non-uniform strut morphology is developed. This model covers several types of cellular structure such as two-dimensional (2D) hexagonal and square honeycombs, and 3D hexagonal and rhombic cellular materials of rod-like columnar structure. The effective elastic constants are determined taking account of bending, axial, and shear deformations of the struts. Unlike the theoretical work of other investigators for 2D honeycombs, considering bending, axial and shearing deformations of struts, the present model not only produces transverse isotropy for regular hexagonal columnar structure but also provides a consistent Poisson's ratio when applied to a square honeycomb. The effect of tapered strut morphology on the elastic properties of cellular structures is investigated. For the general hexagonal columnar structures, the bending compliance is the dominant function for the in-plane elastic constants of 2D and 3D structures (excluding the in-plane shear modulus of rhombic structures) and the out-of-plane shear moduli of 3D structures, but the axial compliance is dominant for the in-plane shear modulus of 2D and 3D rhombic structures and the out-of-plane Young's modulus of 3D structures. For cellular materials with the same relative density, the presence of taper increases values of the effective Young's and shear moduli for which the bending compliance is dominant, but decreases those for which the axial compliance is dominant. It is found that the effective elastic properties of cellular materials are dependent not only on the relative density but also on strut morphology both in cross-section geometry and its variation along the strut length which the present model takes account of. These results illustrate the importance of the strut morphology in calculating the effective elastic properties of cellular materials.     

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.     

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。     

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).     

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.

     

Plane stress-plastic collapse loads for tapered cantilevers and haunched beams

Slip-line fields are proposed for the incipient collapse of tapered cantilevers yielding under combined axial and shear tip loading for plane stress conditions. These are then extended to fixed-ended haunched beams with uniformly distributed or concentrated central loading. Optimization for minimum material use is also considered. All the collapse load results for various taper angles are presented in the form of design curves and some results are compared with those applying for plane strain conditions.     

Failure surfaces for cellular materials under multiaxial loads—II. Comparison of models with experiment

The behaviour of cellular materials under multiaxial loads was modelled in the previous companion paper. Here we present data for the failure of foams by elastic buckling and plastic yielding and compare them to the results of the models. The models describe the main features of the multiaxial behaviour of foams well.     

The effect of axial loading and axial restraint on the thermal ratchetting of thin tubes

The finite element method has been used to investigate the thermal ratchetting behaviour of thin tubes subjected to steady, internal pressure and cyclic, linear, through-thickness temperature distributions. An elastic-perfectly plastic material behaviour model was used and uniaxial behaviour was related to multiaxial behaviour via the von Mises yield criterion and the Prandtl-Reuss flow rule. Results for a tube without axial stress, a tube with axial stress (corresponding to a tube with an end closure) and a tube constrained to have no axial strain, are presented. Correlations with a simplified analytical solution were attempted. Good correlations were obtained for the tubes without axial restraint. The correlation for the axially constrained tube was poor because the thermal loading is essentially different. In the case of the axially constrained tube, if the thermal load is high enough, yielding occurs through the whole of the wall thickness simultaneously in each half cycle. This is not possible in the other two cases considered.     

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。     

考虑附加强化效应及平均应变的多轴疲劳寿命预估

考虑多轴加载条件下裂纹萌生和扩展的物理意义，将Miner理论与临界面法相结合提出了一种多轴等效线性疲劳寿命预估模型；将循环屈服应力与静态屈服应力的比值用来反映材料的循环强化能力，并定义了附加强化因子，考虑相位差及载荷条件对非比例附加强化效应的影响，对非对称加载下的平均应变进行了修正；利用等效应变预估模型、最大剪切应变幅模型及所提模型对5种材料的光滑及缺口试样多轴疲劳寿命进行了预估，并将预估结果与试验结果进行了对比分析，结果表明：比例加载时，3种模型都能获得较好的效果；非比例加载时，所提模型的预估效果要优于其他两种模型的预估效果。同时也验证了所提出的附加强化因子能够反映相位差及材料特性对附加强化效应的影响。     

多轴疲劳寿命预测模型的研究

由于疲劳研究的重要性和多轴疲劳问题存在的普遍性,多轴疲劳理论及其应用的研究逐渐受到了广泛的重视。本文通过比较现有的三种寿命估算方法,由静强度准则引伸出的等效应力（应变）法、能量法和临界平面法,分析了其各自的特点。在试验的基础上,确定了控制多轴疲劳损伤的参量,并以疲劳试验中所获得的数据验证了新建的寿命估算模型。     

Effect of cyclic loading on the yield surface evolution of 18G2A low-alloy steel

Experimental analysis is presented of the plastic properties of 18G2A steel (notation according to Polish Standards) in the as-received state, and of the same material subjected to cyclic predeformation in different directions of the two-dimensional stress space (σ_xx, τ_xy). The analysis was made by studying the position in stress space and by determination of typical dimensions of the yield surfaces. The initial yield surface has been determined using a number of specimens which were loaded up to the plastic range along different stress directions, and this surface was used as the starting point for comparative studies of yield surfaces of the cyclic prestrained material. Cyclic predeformations were induced by loading at ambient temperature. After predeformation, yield surfaces were determined by the technique of sequential probes of the single specimen. The anisotropic yield condition due to Szczepiński was shown to model the experimental results well. Prior cyclic loading induced the softening effect observed during subsequent monotonic loading of the steel in the plastic strain range considered.     

球压痕试验法评价金属材料的强度

通过单次压痕试验与有限元模拟相结合的方法,结合反向分析方法与模拟退火粒子群算法,从获得的载荷-深度曲线加载部分提取材料的塑性参数,基于Ludwig硬化模型预测了不同金属材料的强度,并与单轴拉伸试验结果进行对比。结果表明:模拟得到的载荷-深度曲线与试验得到的几乎重合,二者的相对误差小于0.5%,说明模拟退火粒子群算法可有效地从压痕载荷-深度曲线中提取出金属材料的塑性参数;基于Ludwig硬化模型,利用反向分析方法从压痕载荷-深度曲线中提取的真应力-真塑性应变曲线不是唯一的,但从真应力-真塑性应变曲线计算得到的强度具有明显的收敛趋势;采用压痕试验得到不同金属材料的强度均接近于由拉伸试验得到的,屈服强度与抗拉强度的最大相对误差分别为5.9%,4.3%,说明采用压痕试验法可以准确地评价金属材料的强度。     

A constitutive model for transversely isotropic foams, and its application to the indentation of balsa wood

An elastic-plastic constitutive model for transversely isotropic compressible solids (foams) has been developed. A quadratic yield surface with four parameters and one hardening function is proposed. Associated plastic flow is assumed and the yield surface evolves in a self-similar manner calibrated by the uniaxial compressive (or tensile) response of the cellular solid in the axial direction. All material constants in the model (elastic and plastic) can be determined from a combination of a total of four uniaxial and shear tests. The model is used to predict the indentation response of balsa wood to a conical indenter. For the three cone angles considered in this study, very good agreement is found between the experimental measurements and the finite element (FE) predictions of the transversely isotropic cellular solid model. On the other hand, an isotropic foam model is shown to be inadequate to capture the indentation response.     

锥形流量计下游支撑及取压位置的试验研究

锥形流量计应用广泛,其国际标准正在起草中。锥形流量计发展过程遇到两个问题：安全性和制造一致性差,具有双支撑结构的锥形流量计可解决此问题。为保证增加的双支撑结构不会影响锥形流量计的测量性能和较短的表体长度,需要研究下游支撑和下游取压位置。根据锥形流量计尾流流场的速度分布和压力分布的状况设计相关的改进试验,选择合适的下游支撑和下游取压位置。试验使用的锥形流量计经过特殊设计：锥体、下游支撑位置、下游取压位置都可以按照要求变换位置。试验得到适用于不同直径管道的下游取压位置和支撑位置的选取规律。不同管径和等效直径比下的锥形流量计试验数据显示：改进后的双支撑形式的锥形流量计样机比悬臂梁锥形流量计具有更好的流出系数不确定度。     

变幅单/多轴混合加载下GH4169光滑件的弹塑性有限元分析

对承受变幅单轴和变幅非比例多轴加载的镍基合金GH4169光滑薄壁管进行了有限元分析。使用ANSYS有限元分析软件进行有限元建模,采用Von Mises屈服准则和多线性随动强化准则来描述变幅弹塑性特性。结合三角波加载,在柱坐标系下对试样一端加载轴向位移和周向位移来实现拉扭应变加载。通过对光滑薄壁管件的有限元分析结果与试验结果进行比较,说明此方法对变幅单多轴加载弹塑性分析的正确性和实用性。     

多轴随机应变加载下铝合金缺口件有限元分析及寿命预测

在应变控制下对7075-T7451铝合金实心棒带U型环状缺口试件进行拉—扭比例、非比例恒幅和随机加载疲劳试验。查明名义剪切应力最大值与轴向应力最大值的下降规律,并用两者最先开始下降时的循环数比值来评估裂纹萌生寿命。利用弹塑性有限元法分析不同加载条件下试件缺口根部的循环应力、应变响应。基于有限元计算得到的应力、应变结果,采用拉伸型多轴疲劳损伤模型预测随机加载条件下缺口试件的疲劳裂纹萌生寿命,计算结果与试验得到的寿命比较吻合。     

Plastic deformation modes of regular hexagonal honeycombs under in-plane biaxial compression

A limit analysis approach is employed to identify the plastic deformation modes of regular hexagonal honeycombs with relatively large wall-thickness-to-length ratios under in-plane biaxial compression. An infinite block of honeycomb material is considered and a representative block consisting of four hexagonal cells is defined when assuming the kinematic admissibility of the modes and a periodic repeatability of the representative block in both spatial directions. In general, three plastic collapse modes are found to be preferable depending on the direction of loading, and in some particular cases they are similar to the modes that occur elastically under stress or strain controlled in-plane biaxial compression. It is shown that the critical forces at the onset of the plastic collapse depend on the assumed constraints for the deformation of the representative block. The results obtained from the theoretical analysis and the numerical simulations are compared and discussed.