Elastic moduli and plastic collapse strength of hexagonal honeycombs with plateau borders

The theoretical analysis for the elastic moduli and plastic collapse strength of hexagonal honeycombs with Plateau borders is proposed and presented here. The variation of cell edge thickness in real honeycombs is taken into account in deriving their elastic moduli and plastic collapse strengths. A repeating element, composed of three cell edges connected at a vertex with Plateau borders of constant radius of curvature and width, is employed to calculate the elastic moduli and plastic collapse strength of hexagonal honeycombs. Results suggest that both the elastic moduli and plastic collapse strength of hexagonal honeycombs with Plateau borders depend on their relative density and the volume fraction of solid contained in the Plateau border region. Meanwhile, effects of solid distribution on the elastic moduli and plastic collapse strength of hexagonal honeycombs are investigated, providing a guideline for the optimal microstructure design of honeycombs.     

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.     

In-plane elastic moduli and plastic collapse strength of regular hexagonal honeycombs with dual imperfections

The in-plane elastic modulus, Poisson's ratio and plastic collapse strength of regular hexagonal honeycombs with dual imperfections of non-straight and variable-thickness cell edges were theoretically derived from a model of curved cell edges with Plateau borders. Finite element analyses (FEA) on the stiffness and strength of regular hexagonal honeycombs with dual imperfections were also performed and then compared to the theoretical modeling. Both analytical and numerical results indicate that the in-plane elastic moduli and plastic collapse strength of regular hexagonal honeycombs with dual imperfections depend on their relative density, the solid distribution in cell edges and the curvature of cell edges. Meanwhile, the effects of dual imperfections on the in-plane elastic moduli and plastic collapse strength of regular hexagonal honeycombs are more drastic as compared to those of each single imperfection. Also, it is found that the normalized in-plane elastic modulus and plastic collapse strength of regular hexagonal honeycombs with dual imperfections are approximately equal to the products of those with each single imperfection.     

Effects of concentrated rigid inclusions defects on the in-plane impact behavior of hexagonal honeycombs

Hexagonal honeycombs have exhibited significant advantages in energy absorption and they are increasingly used as absorbers under crush conditions. The in-plane crushing process of imperfect hexagonal honeycombs with concentrated rigid inclusions defects is simulated using finite element simulations. In each case, a constant velocity is applied to an impact plate which then crushed the honeycomb. Simulation results indicate that the defect location has a great influence on the deformation modes, especially at low and moderate velocity. After analyzing the apparent reflection about dynamic response at the impact end, the respective influences of local fraction of inclusions and foil thickness (relative density) on the crushing plateau stress on account of the crushing velocity are further discussed. Furthermore, the energy absorption capacity under constant velocity loading is studied. Due to the distribution of the concentrated rigid inclusions defects, the energy absorption can be controlled effectively.     

The out-of-plane properties of honeycombs

Honeycombs are often used as cores in sandwich panels. The honeycomb cores carry the normal and shear loads in the surfaces perpendicular to the axis of the hexagonal prisms. Honeycombs are particular strong in this out-of-plane direction. In this paper, the collapse behaviour under both shear and simple compression in the out-of-plane direction is analyzed. Buckling, debonding and fracture are identified as possible collapse mechanisms. The modelling work is checked by extensive experiments on a wide range of Nomex honeycombs. Good agreements are found between the model and the data.     

Elastic collapse of honeycombs under out-of-plane pressure

A theoretical scheme is developed to analyze the initial elastic buckling of hexagonal honeycombs with walls of equal or unequal thickness and of square or triangular honeycombs under out-of-plane pressure. The computing results obtained by using this scheme are in good agreement with experimental data.     

Effect of inclusions and holes on the stiffness and strength of honeycombs

A finite element study has been performed on the effects of holes and rigid inclusions on the elastic modulus and yield strength of regular honeycombs under biaxial loading. The focus is on honeycombs that have already been weakened by a small degree of geometrical imperfection, such as a random distribution of fractured cell walls, as these imperfect honeycombs resemble commercially available metallic foams. Hashin–Shtrikman lower and upper bounds and self-consistent estimates of elastic moduli are derived to provide reference solutions to the finite element calculations. It is found that the strength of an imperfect honeycomb is relatively insensitive to the presence of holes and inclusions, consistent with recent experimental observations on commercial aluminium alloy foams.     

Behavior of intact and damaged honeycombs: a finite element study

The Young’s moduli, the elastic buckling strength and the plastic collapse strength of regular honeycombs with defects consisting of missing cells in the structure were analyzed using the finite element method. The behavior of intact honeycombs was first analyzed; the results of this numerical study are consistent with those of previous analyses. The effect of single, isolated defects of varying sizes and the effect of the separation distance between two defects on the elastic and plastic behaviors were then analyzed. Single, isolated defects reduce the modulus and strength. The elastic buckling strength of a honeycomb with a defect normalized by the intact strength decreases directly with the ratio of the minimum net cross-sectional area normalized by the intact cross-sectional area. The plastic collapse strength of a honeycomb with a defect normalized by the intact strength decreases less rapidly than the ratio of the minimum net cross-sectional area normalized by the intact cross-sectional area. Two closely spaced, separate defects interact to reduce the elastic buckling strength of a honeycomb; at a separation distance of about ten cells separate defects act independently. The separation distance between two defects has little effect on the Young’s modulus or the plastic collapse strength of a honeycomb. The finite element analysis allows localization behavior to be studied: we find that the localization strain decreases with increasing .     

Post-collapse characteristics of ductile circular honeycombs under in-plane compression

The post-collapse behaviour of a circular honeycomb material under in-plane compression is analysed in order to estimate the effect of the structural topology on the material strength. A structural approach using the limit analysis and the concept of an equivalent structure is employed to describe the large plastic deformations during post-collapse process. Based on some published experimental results (International Journal of Solids Structures 36 (1999) 4367–996) and our numerical simulations, certain deformation patterns are constructed depending on the direction of loading, and the corresponding post-collapse load-carrying capacities during large deformation until densification of cells are presented.The present analysis shows that the post-collapse stress associated with an equi-biaxial compression is not excessively larger than the corresponding uniaxial stresses, in contrast to those of hexagonal honeycombs in response to biaxial loading. This behaviour is attributed to the different deformation mechanism as the curvature of a cell wall invokes bending without stretching.The influence of the size of the connecting segment between two neighbouring cells is studied, showing that the “shape” of the limit surface varies significantly depending on this connection.     

气动肌肉关节的刚度分析及变刚度运动控制

根据人腿髋关节、膝关节骨骼结构及拮抗肌肉运动发力特点,设计一种拮抗气动肌肉驱动的仿生单腿机器人;由三元素模型求单根肌肉及关节摆动下被动刚度特性,分析关节角度/刚度关系;为实现仿生腿膝关节刚度可控的角度控制,建立仿生关节关于角度/刚度的基本气压解算模型;基于计算力矩控制对非线性对象具有高度补偿线性化性,提出含力矩项补偿的改进气压解算模型。搭建仿真及样机实验平台,结果表明,含两种气压解算模型的双闭环控制算法均能较好跟随膝关节角度/刚度,含带力矩项补偿模型的双闭环控制算法对膝关节的角度/刚度控制精度优于含基本模型的双闭环控制算法。该算法适用拮抗气动肌肉关节的类人运动,可满足人机协作时可靠性、柔顺性、仿生性等要求。     

Effects of defects on the in-plane dynamic crushing of metal honeycombs

The effects of defects and their distributions on the in-plane dynamic crushing of honeycomb panels were studied using explicit finite element modeling. The influence of defect locations and ratios is investigated on the deformation modes and the plateau stresses with respect to the impact velocity. Numerical results show that the dynamic performance of honeycomb displays a high sensitivity on the defect location, especially under low and moderate impact velocities. By introducing a defect correction factor β_m and using the one-dimensional shock wave theory, an empirical formula is given for the variation of honeycomb’s plateau stress with respect to the impact velocity and the defect ratio.     

基于弯曲刚度和扭转刚度的白车身优化分析

随着全球能源的日益紧缺和和制造成本增加,汽车轻量化设计已经成为汽车制造商的主流设计。为了降低白车身的重量,提出了基于刚度灵敏度的方法来实现减重。以处在开发的中后期的某款车为例,利用有限元软件Nastran进行了计算和分析。综合质量灵敏度、刚度灵敏度和优化板件的数量,提出了两种优化方案。并考虑到汽车处在的开发阶段、成本以及整车性能,选取了最佳的优化方案,在不降低汽车性能或者性能降低较小的情况下,实现了车辆的轻量化。最后对优化方案的选用原则和要求进行了总结。     

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.     

Effects of defects on in-plane properties of periodic metal honeycombs

The effects of missing or fractured cell walls on in-plane effective elastic stiffness and initial yield strength of square and triangular cell metal honeycombs are investigated using finite element analysis. Due to the change of localized deformation mode, the in-plane properties of defected honeycombs can differ significantly from those of intact metal honeycombs, depending on cell type and stress state. First, the effect of the size of a statistical volume element of honeycomb cells with randomly removed cell walls is explored by using different numbers of cells with 5% of walls removed, subject to periodic boundary conditions. The size of a representative volume element (statistically homogeneous) is determined for each considered in-plane property. Next, the effective in-plane properties of square cell and triangular cell honeycombs are, respectively, calculated as a function of increasing number density of randomly removed cell walls. Finally, the sensitivities of axial compressive effective properties of these honeycombs to missing cell walls are compared with that of a previously analyzed hexagonal cell honeycomb. The results indicate that some in-plane properties sharply diminish with defect density, while others exhibit more gradual decay. In compression, the effective elastic stiffness and initial yield strength of triangular cell honeycombs are least sensitive to defects among those considered.     

Effects of solid distribution on the elastic buckling of honeycombs

The elastic buckling strengths of honeycombs depend on their relative density, cell geometry and the elastic modulus of solid cell edges. In this study, we consider the effect of the distribution of solid between three cell edges and a vertex on elastic buckling using a semi-analytical integral-equation approach. At first, the geometry of three cell edges connected at a vertex with Plateau borders is analyzed and then employed to represent a repeating element for regular hexagonal honeycombs. The bending moments, rotational angle and the stiffness of a rotational spring corresponding to the constraint from inclined adjacent cell edges are derived for the vertical cell edge within the repeating element. Consequently, the elastic buckling strength of regular hexagonal honeycombs can be numerically obtained. Moreover, the effects of the distribution of the solid on the elastic buckling strengths of regular hexagonal honeycombs are presented and evaluated.     

Improved contact-erosion shaping of diamond wheels and diamond profile grinding

Contact-erosion profiling of diamond wheels based on metallic binder by means of a composite electrode is considered. This method increases the productivity, without reducing the precision, by increasing the profile height of the electrode (tool).     

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.     

The inhomogeneous deformation of polycarbonate circular honeycombs under in-plane compression

In-plane uniaxial and equi-biaxial compression tests were conducted quasi-statically on polycarbonate circular honeycombs. While the uniaxial compression tests were easy to implement, a special test rig was designed to carry out the in-plane (x–y) equi-biaxial compression tests in a conventional universal testing machine. The deformation characteristics of a honeycomb block under uniaxial compression are quantitatively described by tracking the variations of the cells’ parameters, such as the area strain and the angle of cells, during the deformation process of the honeycomb. As the distribution of the deformation within the honeycomb block under biaxial compression is much more complex, a series of color maps based on the area strain of each cell are produced to demonstrate the inhomogeneous deformation among the cells in the honeycomb block, by which the initiation of deformation inhomogeneity and its evolution in the honeycomb block are characterized. To identify the role of the friction between the honeycomb sample and the test rig, finite element analysis is conducted to simulate the collapse process of the circular honeycomb under equi-biaxial compression. Furthermore, an inhomogeneity index, I_inh, is defined as a function of the overall compression of a sample to quantify the severity of the deformation inhomogeneity of the honeycomb; this index facilitates quantitative comparisons among the results under various loading conditions. It is found that the value of I_inh of a sample is closely related to some internal factors, such as the localization band of deformation and the deformation pattern of cells.     

多轴加工系统闭链刚度场建模与刚度性能分析

在复杂曲面多轴加工中,刀具可以在一定范围内以不同姿态加工复杂曲面,刀具姿态的改变直接影响整体加工系统的综合刚度特性,从而影响到加工性能。建立与刀具位姿相关的综合刚度场模型,可用于分析整体加工系统的综合刚性分布规律。提出多轴加工系统刚度场半解析计算方法,其中针对机床运动轴及其传动部位、刀具、工件等关键部件的力学特性分别采用了雅可比矩阵法、点传递矩阵法、有限元法等方法建立相应的刚度矩阵,并根据多体小变形理论建立了多轴加工工艺系统闭链刚度场模型。根据该模型解耦得到三维空间的力椭球,从力椭球提取整体加工系统的刚度性能指标,绘制刚度性能等值线图,分析工作空间中多轴加工系统的刚度特性分布规律,可用于指导刀具运动规划。     

正,负刚度并联结构的稳定性及应用研究

从判定系统静态稳定性的能量准则出发，推导了判定弹性系统静态稳定性的刚度准则，从而十分方法地解决了正、负刚度并联结构的稳定性问题，文中还对此结构在振动控制中的应用进行了研究。