In-plane crushing behavior and energy absorption design of composite honeycombs

Theoretical analysis and numerical simulation methods were used to study the in-plane crushing behavior of single-cell structures and regular and composite honeycombs. Square, hexagonal, and circular honeycombs were selected as honeycomb layers to establish composite honeycomb models in the form of composite structures and realize the complementary advantages of honeycombs with type I and type II structures. The effects of honeycomb layer arrangement, plastic collapse strength, relative density, and crushing velocity on the deformation mode, plateau stress, load uniformity, and energy absorption performance of the composite honeycombs were mainly considered. A semi-empirical formula for plateau stress and energy absorption rate per unit mass for the composite honeycombs was developed. The results showed that the arrangement mode of honeycomb layers is an important factor that affects their mechanical properties. Appropriately selecting the arrangement of honeycomb layers and the proportion of honeycomb layers with different structures in a composite honeycomb can effectively improve its load uniformity and control the magnitude of plateau stress and energy absorption capacity.     

Dynamic crushing behavior of honeycomb structures with irregular cell shapes and non-uniform cell wall thickness

Dynamic crushing responses of honeycomb structures having irregular cell shapes and non-uniform cell wall thickness are studied using the Voronoi tessellation technique and the finite element (FE) method. FE models are constructed for such honeycomb structures based on Voronoi diagrams with different degrees of cell shape irregularity and cell wall thickness non-uniformity. The plateau stress, the densification strain energy and the initiation strain are determined using the FE models. Simulation results reveal that the “X” and “V” shaped deformation modes evident in a perfectly ordered honeycomb at low or moderate impact velocities are disrupted as cell shapes become irregular and/or cell wall thickness gets non-uniform. The “I” shaped deformation mode is clearly seen in all honeycomb structures at high impact velocities. Both the plateau stress and the densification strain energy are found to decrease as the degree of cell shape irregularity or the degree of cell wall thickness non-uniformity increases, with the weakening effect induced by the presence of non-uniform cell wall thickness being more significant. When the two types of imperfections co-exist in a honeycomb structure, the interaction between them is seen to exhibit a complicated pattern and to have a nonlinear effect on both the plateau stress and the densification strain energy. It is also found that stress waves propagate faster in a honeycomb structure having irregular cell shapes and slower in a honeycomb structure having non-uniform cell wall thickness than in a perfectly ordered honeycomb. Finally, the strain hardening of the cell wall material is seen to have a strengthening effect on the plateau stress, which is more significant for perfectly ordered honeycombs than for imperfect honeycomb structures.     

用均匀化理论分析蜂窝结构的等效弹性参数

在线弹性范围内,根据均匀化理论,并结合有限元方法推导出适用于二维周期性结构的均匀化的有限元格式（Homo FEM）,计算出不同相对密度下的规则蜂窝结构的等效弹性模量Ee和泊松比νe.同时,利用蜂窝结构的代表胞元模型,用常规的有限元方法（FEM）模拟计算出相应的等效弹性参数.最后将两种数值计算结果与己有的理论公式进行了比较和分析讨论.结果表明:在考察的相对密度全场范围内（0～0.4）,HOmO FEM得到的蜂窝结构的 Ee和νe 与 FEM使用平面实体单元模拟计算得到的结果一致吻合,反映出 Homo FEM数值方法的客观准确性和可行性.而 Gibson公式和 W-K得到的等效弹性模量值 Ee只是在较小相对密度的情况下（小于0.15）与数值计算结果吻合.当结构的相对密度较大时,必须考虑胞棱附近区域由应力集中导致的复杂的应力和应变分布的影响.     

Buckling and progressive crushing of laterally loaded honeycomb

This paper presents a comprehensive study of the lateral compressive response of hexagonal honeycomb panels from the initial elastic regime to a fully crushed state. Expanded aluminum alloy honeycomb panels with a cell size of 9.53 mm, a relative density of 0.026, and a height of 15.9 mm are laterally compressed quasi statically between rigid platens under displacement control. The cells buckle elastically and collapse at a higher stress due to inelastic action. Deformation then first localizes at mid-height and the cells crush by progressive formation of folds; associated with each fold family is a stress undulation. The response densifies when the whole panel height is consumed by folds. The buckling and crushing events are simulated numerically using finite element models involving periodic domains of a single or several characteristic cells. The models idealize the microstructure as hexagonal, with double walls in one direction. The nonlinear behavior is initiated by elastic buckling while inelastic collapse that leads to the localization observed in the experiments occurs at a significantly higher load. The collapse stress is found to be mildly sensitive to various problem imperfections. The subsequent folding can be reproduced numerically using periodic domains but requires a fine mesh capable of capturing the complexity of the folds. The calculated crushing response is shown to better resemble measured ones when a 4 × 4 cell domain is used. However, the average crushing stress can be captured with engineering accuracy even from a single cell domain.     

Dynamic Crushing Strength Analysis of Auxetic Honeycombs

The in-plane dynamic crushing behavior of re-entrant honeycomb is analyzed and compared with the conventional hexagon topology.Detailed deformation modes along two orthogonal directions are examined,where a parametric study of the effect of impact velocity and cell wall aspect ratio is performed.An analytical formula of the dynamic crushing strength is then deduced based on the periodic collapse mechanism of cell structures.Comparisons with the finite element results validate the effectiveness of the proposed analytical method.Numerical results also reveal higher plateau stress of re-entrant honeycomb over conventional hexagon topology,implying better energy absorption properties.The underlying physical understanding of the results is emphasized,where the auxetic effect(negative Poisson's ratio) induced in the re-entrant topology is believed to be responsible for this superior impact resistance.     

In-plane biaxial crushing of honeycombs—: Part II: Analysis

The in-plane biaxial crushing experiments on polycarbonate honeycomb presented in Part I are simulated using large scale finite element models. The models account for nonlinearities in geometry and due to contact while the polycarbonate is modeled as an elastic-powerlaw viscoplastic solid. Full-scale simulations of the uniaxial crushing of this honeycomb were shown in the past to reproduce experiments with accuracy. In biaxial crushing, it was not practical to model specimens the same size as those in the experiments due to computational limitations; instead, a smaller model with 10×11 cells was adopted. Results from simulations of seven of the crushing experiments in Part I with various biaxiality ratios are presented. Through parametric studies it is demonstrated that the size of the specimen and friction between the specimen and the loading surfaces affect the initial elastic parts of the stress–displacement responses and the onset of instability. By contrast, for average crushing strains higher than approximately 10%, their effect was relatively small and the calculated responses were in good agreement with the experimental ones. As a consequence, the energy absorption capacity was predicted to good accuracy for all biaxiality ratios. In addition, many of the modes of cell collapse seen in the experiment are reproduced in the simulations.     

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.     

Biaxial crushing of honeycombs: —Part 1: Experiments

The in-plane compression and crushing of honeycombs is known to be closely related to the crushing behavior of the broader class of space filling cellular solids. Previously, the authors conducted an extensive study of uniaxial crushing of a polycarbonate honeycomb with circular cells. In this paper the same honeycomb is crushed biaxially. The crushing was performed in a custom testing facility between rigid platens which can be moved independently in two orthogonal directions. The facility allows testing at various biaxiality ratios and volume reductions as high as 95%. The facility was used to conduct several series of biaxial crushing experiments on nearly square honeycomb specimens (18×21 cells) . In each experiment we recorded the true stress–displacement responses in the x- and y-directions as well as full field views of the deformation using a video camera. Biaxial crushing is quite complex and the prevalent mechanisms of collapse depend on the biaxiality ratio (γ) . As is the case in uniaxial crushing, the onset of collapse involves localized instabilities, however, the extent of localized deformation varies with γ. The energy absorption capacity of the material depends on γ. The highest energy is required when the specimen is crushed at the same rates in the two directions.     

正六角形蜂窝夹芯层弯曲刚度理论分析

论文研究了由正六角形胞元构成的正六角形蜂窝夹芯层弯曲刚度.由于正六角形蜂窝夹芯层受面内载荷作用时在胞元斜壁上引起的弯矩与受面外载荷作用时在胞元斜壁上产生的弯矩有所不同,因此,基于面内变形的正六角形蜂窝夹芯层面内等效弹性参数不能用于计算正六角形蜂窝夹芯层弯曲变形.论文基于正六角形蜂窝夹芯层产生弯曲变形时三胞元壁板连接处转角为零的条件会导致正六角形胞元斜壁产生扭曲变形的事实,提出了一种新的正六角形蜂窝夹芯层理论分析方法.用论文方法计算得到的数值结果与有限元商用软件MSC.Marc计算得到的数值解进行对比,表明论文方法不但有效而且可行.     

复合材料襟翼壁板屈曲失稳行为的栅线投影实验研究

本文利用栅线投影测量方法研究了蜂窝夹层板、工字型及T型加筋板三种不同结构形式复合材料襟翼壁板在压缩载荷下的屈曲失稳行为,得到了不同形式结构件屈曲的全场离面位移分布规律,分析了各自的屈曲失稳模式.研究结果表明,栅线投影测量方法在大尺度复合材料结构失稳变形测试中具有可行性;在相同面板尺寸条件下,工字型加筋复合材料襟翼壁板屈曲临界载荷最大,承载能力最强.本文结果可为飞机复合材料结构设计提供实验依据.     

Obtaining a Wide-Strain-Range True Stress–Strain Curve Using the Measurement-In-Neck-Section Method

Background

Measuring true stress–strain curve over a large-strain-range is essential to understand mechanical behavior and simulate non-linear plastic deformation. The digital image correlation (DIC) technique, a non-contact full-field optical measurement technique, is a promising candidate to obtain a long-range true stress–strain curve experimentally.

Objective

This paper proposes a method for measuring true stress–strain curves over a large-strain-range during tensile testing using DIC.

Methods

The wide-strain-range true stress–strain curves of dual-phase and low carbon steels were extracted on the transverse direction in the neck region. The axial strain on the neck section was estimated by averaging the inhomogeneous deformation on the cross-section of the tensile specimen. The true stress was calculated from the engineering stress and the cross-sectional area of the neck.

Results

The validity of the proposed method was assessed by comparing the experimental load–displacement responses during tensile testing with the finite element method (FEM) simulation results. The stress and strain on the neck section estimated using the FEM and DIC, respectively, were proven to satisfy the uniaxial condition and successfully obtained.

Conclusions

The experimental results agree well with the FEM results. The proposed concept can be applied to various deformation modes for accurately measuring long-range true stress–strain curves.

     

Buckling of steel bars with Lüders bands

Experiments and simulations are presented for the study of interaction between material and structural instabilities that occur in mild steel bars under axial compression. The material instability consists of Lüders bands that nucleate and propagate along the specimens. The structural instability involves lateral deflections of the bar leading to collapse. The study includes an investigation of bars of several different lengths. The mechanical response in the experiments were monitored through measurements of axial load, axial and midspan lateral displacements, and full field imaging of a brittle coating showing the Lüders deformation. Interesting interactions exist between the localized deformation due to the material-level instabilities and the global collapse of the bars. Finite element simulations, using a constitutive model with a nonmonotonic stress–strain behavior, showed good agreement with the experiments and helped to explain the variety of collapse modes seen in the experiments. The symmetry of imperfections and/or loading misalignments have dramatic effects on the evolution of Lüders deformation and the eventual collapse mode. Certain imperfections lead to deformation modes that delay structural collapse.     

三维连续与非连续变形分析

将石根华博士所提出的二维非连续变形分析——Discontinuous Deformation Analysis（DDA）方法扩展到三维情况,并对三维不连续块体进行有限元网格剖分,即块体之间的接触采用DDA描述,块体内部的位移场和应力场则采用有限单元法描述,从而将三维DDA与有限元方法结合起来,增强了DDA方法与有限元方法解决实际工程问题的能力,实现了三维连续与非连续变形分析.给出了基本公式的推导过程和各子矩阵的形式.典型接触、碰撞算例证明了所提出方法的有效性和正确性.     

Transient vibration of thin viscoelastic orthotropic plates

This article deals with solutions of transient vibration of a rectangular viscoelastic orthotropic thin 2D plate for particular deformation models according to Flu¨gge and Timoshenko-Mindlin.The linear model,a general standard viscoelastic body,of the rheologic properties of a viscoelastic material was applied.The time and coordinate curves of the basic quantities displacement,rotation,velocity,stress and deformation are compared.The results obtained by an approximate analytic method are compared with numerical results for 3D plate generated by FEM application and with experimental investigation.     

Dynamic modeling and modal truncation approach for a high-speed rotating elastic beam

Summary  In the classical finite element analysis of beams, the nonlinear terms of deformation are ignored due to the linearization of deformation based on the assumptions of structural dynamics. Since the number of generalized coordinates is large in flexible bodies when using the finite element method (FEM), the modal truncation approach (MTA) is usually used for improving computational efficiency, and only lower-order transverse modes are chosen. In this paper, dynamic modeling and application of the MTA to a high-speed rotating beam are studied. The foreshortening displacement is included in the longitudinal displacement, therefore the dynamic modeling takes account of the effect of geometric nonlinearity. Equations of a rotating beam are obtained and the FEM and MTA are used for discretization. The applicability of the MTA to a high-speed rotating elastic beam is verified. The comparison of the results obtained by the FEM and MTA shows that in the case of a high-speed rotation, the centrifugal force can excite high-order transverse modes. Since using lower-order transverse modes for modal truncation obviously can cause error, addition of more transverse modes may improve the result. Furthermore, a coupling effect between axial and transverse displacements is revealed. It is shown that in the case of a sudden change of the axial displacement, the inclusion of the axial modes can significantly improve the response. Received 10 April 2001; accepted for publication 26 March 2002 This work was supported by the National Science Foundation of China (19832040) and the National Education Ministry of China (2000024818), for which the authors are grateful.     

On the application of FEM to deformation of high-density polyethylene

This paper presents a study that uses finite element method (FEM) to simulate deformation behaviour of high-density polyethylene (HDPE) when subjected to tensile loading, either without or with the presence of pre-cracks. For the former, dog-bone-shaped model of HDPE was deformed under uni-axial tensile (UT) loading beyond the initial yielding, to generate stable necking in the gauge section. The paper proposes a simple correction factor to determine the equivalent von Mises stress that is needed for the input to the FEM model, in order to generate the same loading level as that observed experimentally for neck propagation. The paper points out that such consistency in the loading level could not be generated in the past mainly because of a misconception that axial stress in the neck was regarded as the equivalent of the von Mises stress. The study also explored the consideration of crack growth in double-edge-notched tensile (DENT) specimen, and showed that the conventional von Mises yield function, with the assumption of isotropic work hardening, cannot be directly applied to simulate the deformation behaviour of DENT specimen. Instead, empirical parameters were employed to reflect the polymer orientation during the necking process. The paper shows that FEM models for both UT and DENT tests can reproduce the experimental load–displacement curves quite accurately, and concludes that with a proper yield function to reflect the deformation involved in the mechanical tests, the deformation behaviour observed experimentally can be accurately mimicked using the FEM simulation.     

基于板理论的层级褶皱结构失效模式分析

本文基于弹性板理论和夹层板理论对二级层级褶皱结构失效模式进行了分析。通过对基本构件进行受力分析得到了载荷与结构变形之间的关系。根据6种失效模式的定义,从极限载荷或极限应力角度出发,分析了在压缩载荷和剪切载荷工况下的各种失效模式,给出了结构单胞对应的等效正应力和等效切应力表达式。由最小失效强度得到了各失效模式之间的占优关系,并构建了失效机理图来阐释这一机制。最后通过与有限元分析结果比较,分析了本文公式的精度,与数值解吻合较好。     

含随机填充孔圆形蜂窝结构的面内冲击性能

研究多孔材料细观结构与宏观力学性能之间的关系, 建立具有固定相对密度的含随机固体填充孔的圆形蜂窝结构模型。在此模型的基础上具体讨论了不同孔洞填充比和冲击速度对圆形蜂窝结构变形模式、动态冲击平台应力以及能量吸收性能的影响。研究结果表明:填充孔在蜂窝变形过程中有局部牵制作用, 蜂窝材料变形模式仍为准静态模式、过渡模式、动态模式; 当变形模式为过渡模式或动态模式时, 结构的平台应力与速度的平方成线性关系, 存在明显的速度效应; 高速冲击下, 含固体填充孔的蜂窝结构单位质量吸收的能量高于规则蜂窝结构。研究结果可为蜂窝材料的研究和设计提供参考。     

金属蜂窝夹层结构抗水下爆炸特性

金属蜂窝夹层结构是一种新型的舰船防护结构,在舰船防护领域具有广阔的应用前景,但目前缺乏对其在实际水下爆炸载荷作用下动态响应的研究。为研究金属蜂窝夹层结构在水下爆炸载荷作用下的动态响应及防护性能,设计并制备了背板加筋蜂窝夹层结构样件以及相应的浮箱,在大型露天水池中进行了水下实爆 实验;通过声固耦合算法对结构响应进行模拟,实验结果与模拟结果吻合良好,随后分析了蜂窝夹层板的变形过程及能量吸收特性,量化了载荷参数（冲击因子）及结构参数（前后面板厚度比和芯体相对密度）对结构动态响应的影响;最后,以蜂窝夹层板的面密度和后面板中心点最大变形的无量纲量为目标函数,使用NSGA-Ⅱ遗传算法对结构进行了多目标优化,得到对应的Pareto前沿。结果表明,随着冲击因子的增大,蜂窝夹层板整体变形显著增大,蜂窝芯体始终是主要的吸能构件,但其吸能占比逐渐降低;随着前后面板厚度比或芯体相对密度的增加,蜂窝夹层结构的最大变形呈现先降低后升高的趋势,同时呈现不同的变形模式,芯体相对密度对结构变形的影响更为显著;对蜂窝夹层结构开展多目标优化可有效降低结构的面密度及最大变形,优化结果可为蜂窝夹层结构的设计选型提供参考。     

A numerical study on the rate sensitivity of cellular metals

Metallic foams have non-linear deformation behavior, which make them attractive in many applications. Many experimental researches on the dynamic behavior and rate sensitivity of cellular metals have been reported in the literature, but they contain conflicting, and sometimes confusing, conclusions on the strain-rate and inertia effect of cellular metals. In this paper, the dynamic crushing behavior of 2D Voronoi honeycomb is studied by finite element method. The influences of inertia, strain hardening and strain-rate hardening of metal matrix on the deformation mode and plateau stress of the honeycomb are investigated. Three deformation modes are found in different velocity ranges. According to the numerical results, it is found that the plateau stress increases significantly with the increase of impact velocity due to non-uniform deformation induced by inertia. The strain-hardening effect is slight in our numerical tests and the rate effect of the honeycomb is obviously weaker than that of the cell wall material.