A theoretical and experimental study on metal hexagonal honeycomb crushing under quasi-static and low velocity impact loading

In the study of honeycomb crushing under quasi-static loading, two parameters are important; the mean crushing stress and the wavelength of the folding mode. On the other hand, the kinetic energy absorbed by the honeycomb is investigated in the impact loading. In this paper, through fully considering the cylindrical curvature effects and implementing the energy method, a new theoretical model for the estimation of the mean crushing stress and the wavelength of the folding mode of the metal hexagonal honeycomb is presented. Afterwards, developing this static model to the dynamic state, a theoretical model for study of the behavior of these energy absorbers, in the low velocity impact loading, is proposed and the required initial velocity of the impactor, for creation of the desired folding length in these structures, is determined, analytically. The presented theoretical models have been compared with experimental results obtained from experiments on three kinds of honeycomb with the various minor diameters and thicknesses of the cell walls under quasi-static and low velocity impact loading in the axial direction. Excellent correlation has been observed between the theoretical and experimental results.     

Three-Point Bending Behavior of Foam-Filled Conventional and Auxetic 3D-Printed Honeycombs

Cellular hexagonal (conventional) and re-entrant (auxetic) honeycombs are applicable in automotive, construction, and protective engineering. Auxetic structures own excellent energy absorption and flexural behavior due to their special deformation under loading. This work explores the performance of additively manufactured polylactic acid (PLA)- and thermoplastic polyurethane (TPU)-based hexagonal and re-entrant honeycombs under flexural loading via experimental three-point bending (TPB) tests and finite-element analysis (FEA). 3D-printed conventional and auxetic cellular structures are filled with polyurethane (PU) foam and their energy absorption capacity and flexural modulus are compared with hollow structures. The results reveal that TPU-based structures’ energy absorption capacity and flexural modulus improve significantly, whereas the PLA-based structures’ performance deteriorates when filled with PU foam. Moreover, re-entrant honeycombs are better reinforced with foam in comparison to the hexagonal honeycombs, as the re-entrant's unit cell is more spacious than the hexagonal unit cell. Finally, parametric studies are performed via FEA to investigate the influence of geometric parameters of structures and flexural loading setup on the performance of the honeycombs, showing that structures with thicker struts and higher cell angle can act stiffer under TPB. The outcomes of this research indicate the promising performance of foam-filled TPU-based auxetic structures.     

Dynamic energy absorption characteristics of hollow microlattice structures

Hollow microlattice structures are promising candidates for advanced energy absorption and their characteristics under dynamic crushing are explored. The energy absorption can be significantly enhanced by inertial stabilization, shock wave effect and strain rate hardening effect. In this paper we combine theoretical analysis and comprehensive finite element method simulation to decouple the three effects, and then obtain a simple model to predict the overall dynamic effects of hollow microlattice structures. Inertial stabilization originates from the suppression of sudden crushing of the microlattice and its contribution scales with the crushing speed, v. Shock wave effect comes from the discontinuity across the plastic shock wave front during dynamic loading and its contribution scales with v². The strain rate effect increases the effective yield strength upon dynamic deformation and increases the energy absorption density. A mechanism map is established that illustrates the dominance of these three dynamic effects at a range of crushing speeds. Compared with quasi-static loading, the energy absorption capacity at dynamic loading of 250 m/s can be enhanced by an order of magnitude. The study may shed useful insight on designing and optimizing the energy absorption performance of hollow microlattice structures under various dynamic loads.     

Theoretical prediction and numerical simulation of honeycomb structures with various cell specifications under axial loading

The axial crushing of honeycomb structures with various cell specifications is studied analytically and numerically. Based on the Super Folding Element theory, a new method for predicting the mean crushing stress of honeycomb structures with various cell specifications under axial loading is developed. In this new theoretical method, two types of simplified folding modes named SFM1 and SFM2 are proposed. The mean crushing stress and the folding wavelength for honeycomb structures with various cell specifications are then determined by a minimum principle. The effective crushing distance and the loading rate effect are both considered. In order to illustrate the effectiveness of the proposed approach, numerical simulations are carried out by employing the explicit finite element code LS-DYNA. The bond of the honeycomb panels is simulated by using a tie-break contact. It can be seen that the analytical solutions are in agreement with the numerical results as well as the Wizerbicki’s solutions.     

Closed-form solution for crushing response of three-dimensional thin-walled “S” frames with rectangular cross-sections

The crushing behavior of thin-walled rectangular cross-section “S” shaped frames subjected to complex loading cases was analyzed. The results of the biaxial bending collapse of prismatic beams and the collapse under combined bending and compression developed earlier by the present authors were applied to a practical problem of a three-dimensional S frame. The analytical solution of the crushing resistance was derived and the two types of deformation modes were identified using the analysis of the fully plastic bending moment with different orientation angle of bending axis. It was shown that the critical aspect ratio of the rectangular cross-section separating the two deformation modes is 1.366. The analytically derived crushing force gave excellent correlation with the finite element results. The simplified calculation routines derived through this study can be applied to an early design stage of a car body.     

Flexible cellular solid spokes of a non-pneumatic tire

Non-pneumatic tires (NPTs) have been introduced with a compliant cellular solid spoke component which functions as the air of the pneumatic tire. In this paper, hexagonal honeycomb spokes for a high fatigue resistance design are investigated by seeking compliant hexagonal structures that have low local stresses under macroscopic uni-axial loading. Using the honeycomb mechanics, two cases of hexagonal honeycombs are designed: (i) the same cell wall thickness and (ii) the same load carrying capacity. The elastic limits of the hexagonal honeycombs are obtained from the ABAQUS finite element code considering the geometric nonlinearity of a cellular structure associated with the cell wall buckling and bending. The compliant cellular structures having low local stress values are applied to the honeycomb spokes of an NPT for the structural validation and the local stresses of the honeycomb spokes are investigated under the same vertical loading conditions. Hexagonal honeycombs with a highly positive cell angle have low local stresses and low mass under the same vertical load carrying capability; the Type C honeycomb spokes in this study.     

Experimental Characterisation and Numerical Modelling of Hexagonal Honeycomb Cellular Solids under Multi‐axial Loading

Abstract: Cellular solids are becoming increasingly popular for sandwich core and energy‐absorbing applications in many automotive and other transportation structures. This paper investigates experimentally and numerically the strength and post‐failure energy absorption of a popular hexagonal aluminium honeycomb material under multi‐axial loading conditions. For the experimental work, an improved Arcan test apparatus is used so that interaction of multi‐axial compression and shear loading on failure and crushing may be studied; optical measuring methods are used to extract deformation data. In addition, experimental work to characterise the material with pre‐deformation in the in‐plane directions has also been conducted. This experimental work provides input for computational modelling of the material and two alternative modelling approaches have been investigated. First, a three‐dimensional anisotropic, elastic–plastic model, with coupling of loading components is used to represent the material at the macro‐level and, second, a meso‐modelling approach using a fine shell representation of the thin‐walled honeycomb cellular structure is applied. For practical analysis of large‐scale structures, the former approach is computationally efficient and can reasonably treat the most important failure and crush characteristics of the material. However, for more accurate analysis, particularly in the case of complex non‐proportional loading, the meso‐shell model may provide a more realistic solution.     

Dynamic crushing behavior of 3D closed-cell foams based on Voronoi random model

Dynamic crushing responses of three-dimensional cellular foams are investigated using the Voronoi tessellation technique and the finite element (FE) method. FE models are constructed for such closed-cell foam structures based on Voronoi diagrams. The plateau stress and the densification strain energy are determined using the FE models. The effects of the cell shape irregularity, impact loading, relative density and strain hardening on the deformation mode and the plateau stress are studied. The results indicate that both the plateau stress and the densification strain energy can be improved by increasing the degree of cell shape irregularity. It is also found that the plastic deformation bands appear firstly in the middle of the model based on tetrakaidecahedron at low impact velocities. However, the crushing bands are seen to be randomly distributed in the model based on Voronoi tessellation. At high impact velocities, the “I” shaped deformation mode is clearly observed in all foam structures. Finally, the capacity of foams absorbing energy can be improved by increasing appropriately the degree of cell shape irregularity.     

Elasto-plastic strain analysis by a semi-analytical method

The aim of this paper is to develop a simulation model of large deformation problems following a semi-analytical method, incorporating the complications of geometric and material non-linearity in the formulation. The solution algorithm is based on the method of energy principle in structural mechanics, as applicable for conservative systems. A one-dimensional solid circular bar problem has been solved in post-elastic range assuming linear elastic, linear strain hardening material behaviour. Type of loading includes uniform uniaxial loading and gravity loading due to body force, whereas the geometry of the bar is considered to be non-uniformly taper. Results are validated successfully with benchmark solution and some new results have also been reported. The location of initiation of elasto-plastic front and its growth are found to be functions of geometry of the bar and loading conditions. Some indicative results have been presented for static and dynamic problems and the solution methodology developed for one-dimension has been extended to the elasto-plastic analysis of two-dimensional strain field problems of a rotating disk.     

倾斜胞壁Nomex蜂窝芯压剪复合力学响应

成形具有一定曲率的夹层结构时,需要将蜂窝芯铣削成曲面形状,造成蜂窝胞壁呈一定倾角,进而降低蜂窝夹芯结构面外承载能力。为了定量化分析面外载荷作用下倾斜胞壁蜂窝芯的力学性能,建立了倾斜胞壁蜂窝芯面外压剪复合有限元模型,并通过设计专用Arcan夹具实现蜂窝芯的面外压剪复合加载,用于验证模型的有效性。对比仿真与实验结果,发现蜂窝芯压剪响应及胞壁变形模式吻合较好。利用验证的有限元模型对胞壁倾角范围为0°～40°的蜂窝芯在面外压剪复合载荷下的力学响应进行了研究,结果表明随着蜂窝胞壁倾角的增大,蜂窝芯面外承载能力逐渐降低;当胞壁倾斜角由0°增加到40°,初始应力峰值下降最大幅度为47.7%,平原阶段强度下降幅度为29%;进一步分析了倾斜胞壁蜂窝芯截面芯格尺寸与胞壁倾角的几何关系,将倾斜胞壁蜂窝芯等效为具有相同截面尺寸的垂直胞壁蜂窝芯,推导了倾斜胞壁蜂窝芯在面外压缩及剪切载荷作用下的坍塌强度,揭示了胞壁倾角对蜂窝芯坍塌强度影响机制。     

组合蜂窝材料面内冲击性能的研究

基于三角形和六边形蜂窝结构面内冲击性能的研究,该文探讨了面内冲击荷载作用下组合Kagome蜂窝结构的变形机制和能量吸收特性。首先,在保证蜂窝结构胞元厚度与边长尺寸比值不变的前提下,分析了不同形状胞元及其组合结构的动态冲击性能,给出了试件宏观及微观胞元结构的动态演化过程。在此基础上,探讨了冲击速度和相对密度一定情况下单位质量不同蜂窝结构的能量吸收特性。其结论将对蜂窝材料微拓扑结构的动力学优化设计提供指导。     

Crushing behavior of hybrid hexagonal/octagonal cellular composite system: Aramid/carbon hybrid composite

In the current paper a series of experiments were conducted to assess the crashworthiness of cellular hexagonal/octagonal composite device. Each device composed of 6 cells of carbon fiber reinforced composite (CFRP). Different arrangements of the octagonal and the hexagonal cells were studied. All the configurations were filled with foam. The main objective of the current paper was to examine the effect of using the aramid/epoxy instead of the carbon/epoxy layers to pack the device. The specimens were tested under quasi-static compression loading up to complete crushing. The results showed that the packing material did not have a significant effect for the case of all hexagonal open cells. For the other configurations, introducing the aramid/epoxy instead of the carbon/epoxy showed improvements in the stroke efficiency, the crush load stability, the average crushing load, the energy absorbed and the specific energy absorption. In order to understand the mechanisms that led to this improvement, the packing material were examine after crushing using an optical microscope and a scanning electron microscope (SEM). For the carbon/epoxy, the images showed many failure mechanisms whereas, for the aramid/epoxy, only delamination was noted.     

超薄壁钢筋混凝土水塔的爆破拆除

介绍了爆破拆除一薄壁钢筋混凝土水塔的实例。针对薄壁这一特点 ,利用分析法计算爆破切口长度。定向口采用组合形式 ,以增大爆高。从炸药能量主要用于介质的剪切、破碎的角度 ,计算单孔装药量。这些见解可供类似结构的爆破拆除作参考     

Analysis of debonding fracture in a sandwich plate with hexagonal core

In lightweight applications (as, e.g., aerospace structures) sandwich constructions are very useful and common due to their superior specific bending stiffness and bending strength. In many cases the sandwich consists of an upper and lower laminate facesheet and an intermediate hexagonal cellular aluminum core. Along their interfaces the facesheets and the core are glued together. In order to ensure structural integrity, the facesheet/core bonding is of particular interest. Finite element method has been used to study the cause and the effects of debonding phenomena in between the facesheet and the core of a sandwich plate under in-plane loading. A “unit cell” approach has been followed throughout the study. It has been observed that under an applied in-plane loading, there is a significant stress concentration at the junction of three cell walls and facesheet which easily leads to the generation of cracks and their growth. In order to judge about the tendency of crack initiation and growth, hypothetical interface cracks have been considered and analyzed by fracture mechanics technique. In doing so for various crack length, the energy release rate has been calculated and assessed by means of Irwin’s crack closure integral for a number of different situations. It has been observed that there is a significant amount of energy release rate even in the case of a very small or virtually no crack. This phenomenon indicates that the glue used to attach the facesheet and the cell must withstand a non-zero energy release rate even in the intact situation without any debonding.     

Second-order cone programming with warm start for elastoplastic analysis with von Mises yield criterion

The incremental problem for quasistatic elastoplastic analysis with the von?Mises yield criterion is discussed within the framework of the second-order cone programming (SOCP). We show that the associated flow rule under the von?Mises yield criterion with the linear isotropic/kinematic hardening is equivalently rewritten as a second-order cone complementarity problem. The minimization problems of the potential energy and the complementary energy for incremental analysis are then formulated as the primal-dual pair of SOCP problems, which can be solved with a primal-dual interior-point method. To enhance numerical performance of tracing an equilibrium path, we propose a warm-start strategy for a primal-dual interior-point method based on the primal-dual penalty method. In this warm-start strategy, we solve a penalized SOCP problem to find the equilibrium solution at the current loading step. An advanced initial point for solving this penalized SOCP problem is defined by using information of the solution at the previous loading step.     

On a numerical solution of the plastic buckling problem of structures

An automated digital computer procedure is presented for the accurate and efficient solution of the plastic buckling problem of structures. This is achieved by a Sturm sequence method employing a bisection strategy, which eliminates the need for having to solve the buckling eigenvalue problem at each incremental (decremental) loading stage that is associated with the usual solution techniques. The plastic bucking mode shape is determined by a simple inverse iteration process, once the buckling load has been established. Numerical results are presented for plate problems with various edge conditions. The resulting computer program written in FORTRAN for the JPL UNIVAC 1108 machine proves to be most economical in comparison with other existing methods of such analysis.     

Dynamic response of a multilayer prismatic structure to impulsive loads incident from water

The dynamic crush response of a low relative density, multilayered corrugated core is investigated by combining insights from experiments and 3D finite element simulations. The test structures have been fabricated from 304 stainless steel corrugations with 0°/90° lay-up orientation and bonded by means of a transient liquid phase method. Characterization of the dynamic crushing of these structures has revealed that at low rates, interlayer interactions induce a buckling-dominated soft response. This softness is diminished at high rates by inertial stabilization and the response of the structure transitions to yield-dominated behavior. Unidirectional dynamic crushing experiments conducted using a dynamic test facility reveal a soft response, consistent with lower rate crushing mechanisms. The 3D simulation predictions of crushing strain, pulse amplitude/duration and impulse delivery rate correspond closely with the measurements. The application of core homogenization schemes has revealed that by calibrating with a multilayer unit cell, high fidelity continuum level predictions are possible. Moreover, even simplified hardening curves based on equivalent energy absorption provide remarkably accurate predictions of the crush strains and the impulse transmitted through the core. The multilayered structures investigated here significantly reduced the transmitted pressures of an impulsive load.     

膜结构极小曲面找形的一种自适应有限元分析

找形分析是膜结构设计中的关键环节,但在数学上,膜结构的极小曲面找形分析是一个高度非线性问题,一般无法求得其解析解,因此数值方法成为重要工具。近年来,基于单元能量投影法（EEP法）的一维非线性有限元的自适应分析已经取得成功,基于EEP法的二维线性有限元自适应分析也被证实是有效、可靠的。在此基础上,该文提出一种基于EEP法的二维非线性有限元自适应方法,并成功将之应用于膜结构的找形分析。其主要思想是,通过将非线性问题用Newton法线性化,引入现有的二维线性问题的自适应求解技术,进而实现二维有限元自适应分析技术从线性到非线性的跨越,将非线性有限元的自适应分析求解从一维问题拓展到二维问题。该方法兼顾求解的精度和效率,对网格自适应地进行调整,最终得到优化的网格,其解答可按最大模度量逐点满足用户设定的误差限。该文综述介绍了这一进展,并给出数值算例用以表明该方法的可行性和可靠性。     

Bending Crush Resistance of Partially Foam‐Filled Sections

Potential applications of foam‐filled sections are for the automotive structures. A foam‐filled section can be used for the front rail and firewall structures to absorb impact energy during frontal or side collision. In the case of biaxial loading where bending and axial compression are involved in the crushing mechanics, the foam filler will be significant in maintaining progressive crushing of the thin‐walled structures so that more impact energy can be absorbed. In the case of side collision, the foam‐filled section can be used to strengthen the B‐pillar structure to avoid severe intrusion in the passenger compartment.     

Multiobjective optimization of multi-cell sections for the crashworthiness design

Plastic deformation of structures absorbs substantial kinetic energy when impact occurs. For this reason, energy-absorbing components have been extensively used in the structural design of vehicles to intentionally absorb a large portion of crash energy to reduce the severe injury of occupants. On the other hand, high peak crushing force may to a certain extent indicate the risk of structural integrity and biomechanical damage of occupants. For this reason, it is of great significance to maximize the energy absorption and minimize the peak force by seeking for optimal design of these components. This paper aims to design the multi-cell cross-sectional thin-walled columns with these two crashworthiness criteria. An explicit finite element analysis (FEA) is used to derive higher-order response surfaces for these two objectives. Both the single-objective and multi-objective optimizations are performed for the single, double, triple and quadruple cell sectional columns under longitudinal impact loading. A comparative analysis is consequently given to explore the relationship between these two design criteria with the different optimization formulations.