Regular Hexagonal Honeycomb Research Articles

Elastic Wave Propagation in Hierarchical Honeycombs With Woodpile-Like Vertexes

This paper investigates the dispersion behavior of elastic wave propagation in hierarchical honeycombs using the finite element method in conjunction with the Bloch's theorem. The hierarchical honeycomb is constructed by replacing each vertex of a regular hexagonal honeycomb with smaller hexagons stacked in a woodpile pattern. Band structure analysis reveals that, in the considered range of frequency, the maximum band gap for the hierarchical honeycomb is localized in the frequency corresponding to the natural vibration frequency of the cell strut, and moreover, the width of this particular gap is significantly broadened as the order of hierarchy increases. In addition, for the hierarchical honeycombs satisfying an invariable ratio between the thickness and squared length of the cell strut, which is extracted from the expression of the natural frequency of the simply supported element beam, a coincidence among dispersion curves (or contours) for the hierarchical configurations with the same scale order occurs. The resulting identical band gaps as well as the quasi-static phase wave velocities provide an advantage or the hierarchical honeycombs in the manipulation of vibration and associated multifunction designs.

Enhanced tunable fracture properties of the high stiffness hierarchical honeycombs with stochastic Voronoi substructures

Abstract For multifunctional optimization design of honeycomb structures, the high stiffness hierarchical honeycombs with stochastic Voronoi substructures (HHSVS) are proposed by substituting cell walls of the regular hexagonal honeycombs (ORHH) with Voronoi honeycomb lattices of equal mass. In this study, the tunable linear elastic fracture properties of the HHSVS are investigated by finite element analysis. Results demonstrate that size effect on fracture toughness of the HHSVS is noticeable and a brittleness number is suggested to determine it. At the same time, compared with the ORHH and conventional Voronoi honeycombs of equal mass/density, the in-plane fracture toughness of the HHSVS could be more than 2 times larger and regardless of the cell regularity and hierarchical parameters, fracture toughness of the HHSVS exhibits a weaker quadratic dependence on the relative density and is the highest. As a whole, the HHSVS exhibit the combined properties of tunable Poisson’s ratio, higher stiffness, enhanced tunable fracture toughness, lower imperfection sensitivity and better structural-acoustic performance etc. The research provides a novel strategy for the multifunctional optimization design of the honeycombs structures widely used in the engineering fields.

Enhanced strength and weakened dynamic sensitivity of honeycombs by parallel design

This paper proposes a parallel design which can extremely enhance the in-plane property of honeycombs without adding extra mass. Theoretical analysis of regular hexagonal honeycomb with parallel gradient configuration (HPGC) demonstrates that it can achieve a maximum of more than 70% enhancement of static strength in the x direction compared with uniform honeycomb (UH) of the same relative density. Furthermore, static enhancement coefficient is irrelevant to the average relative density of HPGC, which means the degree of enhancement under certain density gradient is constant no matter what the relative density is employed. This is efficient for fast performance-oriented design. The mechanism of strength enhancement is discussed. Result shows that parallel design is applicable to many cellular solids who are eligible for the special relations (for example, the linear relation between graded variable t and relative density and the nonlinear relation between graded variable t and crushing strength). Dynamic analysis shows that the dynamic crushing strength is also enhanced by parallel design. In addition, parallel design is able to reduce the sensitivity of dynamic crushing strength to the impact velocity. Finite element analyses (FEA) are carried out to verify the theoretical results. This novel design provides huge potentials to cellular solids with high-level strength and lightweight applications.

Buckling of honeycomb structures under out-of-plane loads

The governing equations for the buckling of honeycomb cores with various cell geometries under combined compression and shear are established and three types of core including rectangular, hexagonal and triangular cores are under consideration. After invoking the Bloch wave representation form, the equations are simplified by the periodicity and the hypothesis that the out-of-plane displacement remains zero at the intersections. Different cell geometries and load cases are taken into account and numerical results offer validation for the analytical solutions. Moreover, the results of Finite Element (FE) models show that the fine results can only be acquired by models with appropriate cell numbers. Experimental study is conducted on the regular hexagonal honeycomb structures. Both the results of the numerical benchmarks and the experiments prove the effectiveness of the proposed analytical method and the hypothesis for predicting the buckling load of honeycomb structures.

Experimental and Numerical Study of Honeycomb Tip on Suppressing Tip Leakage Flow in Turbine Cascade

In this paper, the effect of a novel honeycomb tip on suppressing tip leakage flow in turbine cascade has been experimentally and numerically studied. Compared to the flat tip cascade with 1%H blade height, the relative leakage flow in honeycomb tip cascade reduces from 3.05% to 2.73%, and the loss also decreases by 8.24%. For honeycomb tip, a number of small vortices are rolled up in the regular hexagonal honeycomb cavities to dissipate the kinetic energy of the clearance flow, and the fluid flowing into and out the cavities create aerodynamic interceptions to the upper clearance flow. As a result, the flow resistance in the clearance increased and the velocity of leakage flow reduced. As the gap height increases, the tip leakage flow and loss changes proportionally, but the growth rate in the honeycomb tip cascade is smaller. Considering its wear resistance of the honeycomb seal, a smaller gap height is allowed in the cascade with honeycomb tip, and that means honeycomb tip has better effect on suppressing leakage flow. Part honeycomb tip structure also retains the effect of suppressing leakage flow. It shows that locally convex honeycomb tip has better suppressing leakage flow effect than the whole honeycomb tip, but locally concave honeycomb tip is slightly less effective.

Numerical and analytical investigation on crushing of fractal-like honeycombs with self-similar hierarchy

The out-of-plane crushing resistance of fractal-like honeycombs with self-similar hierarchy are investigated numerically and analytically. The hierarchical honeycombs are developed via replacing each three-edge vertex of a regular hexagonal honeycomb by a smaller hexagon. Hierarchical honeycombs with higher orders are constructed by repeating this process. Theoretical solutions are derived based on the Super Folding Element (SFE) theory to predict the mean crushing forces (MCFs) of the hierarchical honeycombs. Numerical simulations are also conducted in conjunction with the theoretical solutions. Crushing resistances of three groups of hierarchical honeycombs with different relative densities are explored subsequently. Good agreement between results obtained using theoretical and numerical methods indicates that the theoretical predictions are reliable. The results show that hierarchy can significantly improve the crushing resistance of honeycombs. It is found that the MCFs of first to fourth order hierarchical honeycombs are improved by 71%, 114%, 201% and 309% compared with the regular honeycomb under the same relative density, respectively. It reveals that the amplification of the MCF from one generation to the next approaches a constant (ξn ≈ 1.26) when the hierarchical order is sufficiently large. The potential maximum achievable amplifications of the MCF for hierarchical honeycombs of different relative densities are also discussed.

Out-of-plane crashworthiness analysis of bio-inspired aluminum honeycomb patterned with horseshoe mesostructure

Numerous composite structures with excellent integrative performance that can replicate the mechanical properties of biological materials have been created to fill gaps in material-property charts, and these bio-inspired structures have crucial implications in a wide range of engineering communities. In this paper, a series of novel bio-inspired aluminum honeycombs consisting of horseshoe mesostructure have been proposed on the basis of triangular honeycomb, square honeycomb, hexagonal honeycomb and kagome honeycomb to improve the energy absorption capacity. The three-dimensional finite element models of the bio-inspired horseshoe-shaped aluminum honeycombs are developed in order to explore the mechanical behaviors under the out-of-plane uniform compression. The simulation results are validated based on the compression experiments of regular hexagonal honeycombs. Besides, parametric investigations are carried out to understand the influences of the wave amplitude, wave number and cell-wall thickness on the out-of-plane crashworthiness. The numerical results demonstrate that adding the horseshoe mesostructure to the regular honeycombs can increase the plateau force greatly compared with the traditional honeycomb structure, leading to the higher specific energy absorption although increasing the initial peak force as well. Finally, a multi-objective optimization is carried out to seek for the optimal honeycombs with the maximum specific energy absorption together with the minimum initial peak force simultaneously.

Effective In-Plane Elastic Modulus of a Periodic Regular Hexagonal Honeycomb Core with Thick Walls

AbstractThe effective in-plane elastic moduli of different types of honeycomb cores with thick walls are studied using analytical and numerical approaches. Analytical equations derived in this pape...

A numerical approach to study the post-yield softening in cellular solids: role of microstructural ordering and cell size distribution

Designing meta-materials and cellular solids with biomimetic structures has received increasing attention in the past few years partially due to advances in additive manufacturing techniques that have enabled the fabrication of advanced materials with arbitrarily complex microarchitectures and novel functionalities. To impact on this trend, it is essential to develop our understanding about the role of microstructure on mechanical responses of these structures. Although a large literature exists on the general subject, the role of microstructure on the post-yield instability is not yet adequately documented. This research introduces a numerical approach to study the post-yield instability in 2D infinite honeycombs as a bottleneck for understanding the instability in more complex 3D systems. Two distinct algorithms are used to systematically vary the ordering in the microstructure of regular hexagonal honeycombs and the mean cell size in the microstructure of Voronoi honeycombs. Finite elements together with a nonlinear homogenization technique are incorporated to quantify the instability responses. Finally, the contribution of microstructure on initial post-yield instability—the slope and magnitude of stress drop after yielding—is statistically investigated. It is found that changing the degree of ordering in the microstructure with respect to a parent symmetry can systematically vary the post-yield instability properties. However, systematic variation of the cell size distribution in absence of ordering in the microstructure cannot remarkably change the post-yield instability in the meta-material. Instead, a tailored cell size distribution can systematically influence its yield strength and plateau stress.

Behavior Analysis of Education Robot Scenario Based on WiFi Fingerprint Indoor Positioning Environment

Communication positioning has a significant application in modern life. The positioning of communication terminals is divided into two categories: outdoor localization and indoor localization. At present, many researchers are limited to the pursuit of positioning accuracy, so they rarely carry out a comprehensive study, such as the positioning objects, environmental construction and data acquisition etc., which leads to a reduction in the value of the applications. In the paper, our research object is based on the education robot, whose active scenario is indoor, using WiFi fingerprint positioning technology, Aps layout and RSS data acquisition and process under the offline environment, which facilitates the further research about online stage in the later. In the study, the APs were arranged in a regular hexagonal honeycomb, rational use and clustering signals, which solves the strong interference between APs and improves the usability of the signals.

Mechanical properties of additively manufactured octagonal honeycombs

Honeycomb structures have found numerous applications as structural and biomedical materials due to their favourable properties such as low weight, high stiffness, and porosity. Application of additive manufacturing and 3D printing techniques allows for manufacturing of honeycombs with arbitrary shape and wall thickness, opening the way for optimizing the mechanical and physical properties for specific applications. In this study, the mechanical properties of honeycomb structures with a new geometry, called octagonal honeycomb, were investigated using analytical, numerical, and experimental approaches. An additive manufacturing technique, namely fused deposition modelling, was used to fabricate the honeycomb from polylactic acid (PLA). The honeycombs structures were then mechanically tested under compression and the mechanical properties of the structures were determined. In addition, the Euler-Bernoulli and Timoshenko beam theories were used for deriving analytical relationships for elastic modulus, yield stress, Poisson's ratio, and buckling stress of this new design of honeycomb structures. Finite element models were also created to analyse the mechanical behaviour of the honeycombs computationally. The analytical solutions obtained using Timoshenko beam theory were close to computational results in terms of elastic modulus, Poisson's ratio and yield stress, especially for relative densities smaller than 25%. The analytical solutions based on the Timoshenko analytical solution and the computational results were in good agreement with experimental observations. Finally, the elastic properties of the proposed honeycomb structure were compared to those of other honeycomb structures such as square, triangular, hexagonal, mixed, diamond, and Kagome. The octagonal honeycomb showed yield stress and elastic modulus values very close to those of regular hexagonal honeycombs and lower than the other considered honeycombs.

In-plane crushing of a hierarchical honeycomb

• The crushing of a second order hierarchical honeycomb is investigated. • The failure modes for low and high velocity impacts are revealed by simulations. • A two-scale method is proposed to obtain the analytical collapse stress. • The hierarchical honeycomb has an improved collapse stress. Properly introduced hierarchy in cellular materials has the potential to further improve their energy absorption capacity. The in-plane uniaxial collapse response of a second order hierarchical honeycomb (i.e., a regular hexagonal honeycomb with its cell walls consisting of an equilateral triangular honeycomb) is investigated. Its failure modes for quasi-static crushing and dynamic impact in two directions are systematically explored by finite element simulations. A two-scale method is proposed and analytical expressions for the quasi-static collapse stresses of the hierarchical honeycomb in the two directions are obtained. In conjunction with the conservation of momentum, the analytical quasi-static collapse stress models are extended to dynamic crushing. The obtained theoretical collapse stresses are validated by finite element simulations for a wide range of impact velocity and relative density. Both numerical and analytical results show that the hierarchical honeycomb has an improved collapse stress over traditional hexagonal and triangular honeycombs. The improvement is found to be more pronounced for low velocity impact than for high velocity impact.

Out-of-plane crashworthiness of bio-inspired self-similar regular hierarchical honeycombs

Hierarchical materials are widely observed in nature, and have been credited with superior mechanical properties and weight efficiency. In the present work, fractal-appearing self-similar regular hexagonal hierarchical honeycombs (HHHs) are constructed by iteratively replacing each three-edge vertex of a base hexagonal network with a smaller regular hexagon up to second order. The cell wall thickness is adjusted so that the two fractal configurations have identical density as the base honeycomb. To investigate the out-of-plane crashworthiness of this new class of hexagonal hierarchical honeycomb concept, finite element modeling is carried out and validated using experiment results. By comparing the results of three traditional corrugated hexagonal honeycombs (0th order HHH) with those pertinent to two novel 1st and 2nd order HHHs, it can be concluded that hierarchical organization of different cells can enhance the material/strength distribution across the network, resulting in improved crush strength and crush force efficiency. In addition, parametric studies are carried out to explore two further strategies to improve out-of-plane crashworthiness by altering the material distribution, namely changing relative cell sizes and cell wall thickness. The results suggest that further improvement can be realized through optimum designs of the fractal geometries.

Micropolar Constitutive Relations for Cellular Solids

With the recent development of micromechanics in micropolar solids, it is now possible to characterize the macroscopic mechanical behavior of cellular solids as a micropolar continuum. The aim of the present article is to apply these methods to determine the micropolar constitutive relation of various cellular solids. The main focus will be on the hexagonal packed circular honeycomb to demonstrate how its constitutive relationship is obtained. In addition, the same method will be applied to determine the material properties of a grid structure and a regular hexagon honeycomb. Since we model the cellular solid as an assembly of Euler–Bernoulli beams, we know that the macroscopic material properties will depend on the cell wall thickness, length, and Young's modulus. From this, and in conjunction with nondimensional analysis, we can provide a closed form solution, up to a multiplicative constant, without resorting to analyzing the governing equations. The multiplicative constant is evaluated through a single numerical simulation. The predicted values are then compared against assemblies with different local properties, using the numerical result as a benchmark since it takes into account higher order thickness effects. It is concluded that our closed form expressions vary from the numerical predictions only when the thickness of the beams increase, as expected since shear effects must be taken into account. However, for most engineering applications, these expressions are practical since our closed form solution with the Euler–Bernoulli assumption only produces about 10% error for most extreme cases. Our results are also verified by comparing them against those reported in the literature.

Inserting Stress Analysis of Combined Hexagonal Aluminum Honeycombs

Two kinds of hexagonal aluminum honeycombs are tested to study their out-of-plane crushing behavior. In the tests, honeycomb samples, including single hexagonal aluminum honeycomb (SHAH) samples and two stack-up combined hexagonal aluminum honeycombs (CHAH) samples, are compressed at a fixed quasistatic loading rate. The results show that the inserting process of CHAH can erase the initial peak stress that occurred in SHAH. Meanwhile, energy-absorbing property of combined honeycomb samples is more beneficial than the one of single honeycomb sample with the same thickness if the two types of honeycomb samples are completely crushed. Then, the applicability of the existing theoretical model for single hexagonal honeycomb is discussed, and an area equivalent method is proposed to calculate the crushing stress for nearly regular hexagonal honeycombs. Furthermore, a semiempirical formula is proposed to calculate the inserting plateau stress of two stack-up CHAH, in which structural parameters and mechanics properties of base material are concerned. The results show that the predicted stresses of three kinds of two stack-up combined honeycombs are in good agreement with the experimental data. Based on this study, stress-displacement curve of aluminum honeycombs can be designed in detail, which is very beneficial to optimize the energy-absorbing structures in engineering fields.

Mechanical properties of the hierarchical honeycombs with stochastic Voronoi sub-structures

The introduction of hierarchy into structures has been credited with changing mechanical properties. In this study, periodically hierarchical honeycomb with irregular sub-structure cells has been designed based on the Voronoi tessellation algorithm. Numerical investigation has been performed to determine the influence of structural hierarchy and irregularity on the in-plane elastic properties. Irregular hierarchical honeycombs can be up to 3 times stiffer than regular hexagonal honeycombs on an equal density basis. Both the stiffness and Poisson's ratio of the hierarchical honeycomb are insensitive to the degree of regularity, and depend on the cell-wall thickness-to-length ratio of the super-structure. Increasing the relative lengths of the super- and sub-structures results in the increment of Young's modulus, whereas Poisson's ratio almost remains constant varying from 1.0 to 0.7.

Parametric study and multi-objective crashworthiness optimisation of reinforced hexagonal honeycomb under dynamic loadings

Hexagonal honeycombs have exhibited significant advantages in energy absorption and they are increasingly used as absorbers under crush conditions. Rather than restating details of the existing traditional regular hexagonal honeycomb, this paper introduces reinforced hexagonal honeycomb. The full-scale elaborator finite-element model created through LS-DYNA is validated by available theoretical solutions and experimental results. Afterwards, the validated simulation model is further applied to do a number of parametric studies on the reinforced hexagonal honeycomb under dynamic impact. The parameters such as stiffener thickness, expanding angle, cell length, cell wall thickness, impact mass, impact velocity and end constraint are employed to determine their relative effects on crushing behaviour of the reinforced hexagonal honeycomb. In the multi-objective crashworthiness optimisation, optimal Latin hypercube design method is used to select the sampling design points from the design space. To obtain a maximum specific energy absorption per mass ( ) capability and minimum initial peak stress ( ), response surface method, which is an accurate surrogate modelling method is adopted. The optimisation results show that the reinforced hexagonal honeycomb has a better energy absorption performance within the same limit of . This research is hence significant in providing technical support in potential applications of the reinforced hexagonal honeycomb used as crashworthiness structures.

Crushing analysis and crashworthiness optimization design of reinforced regular hexagon honeycomb sandwich panel

Abstract Honeycomb sandwich panels (HSP) have exhibited significant advantages in light weight and energy absorption, and they have been widely applied in the automotive, aerospace, transportation and defense industries. Rather than restating details of the existing standard regular hexagon HSP (R0-HSP), this paper introduces single-rib reinforced regular hexagon HSP (R1-HSP) and double-rib reinforced regular hexagon HSP (R2-HSP). The mechanical characteristics of these three structures are first investigated by series full-scale elaborator models through LS-DYNA. It is found that the R1-HSP has the best crashworthiness performance under axial impact regarding specific energy absorption per unit (SEAm and SEAv) when the peak crashing stress (σ peak) is less important. Secondly, the thickness matching effect between stiffener and cell wall is analyzed with equivalent apparent density index to evaluate the contribution of the change of deformation model to the general mechanical properties. Consequently, stiffener thickness of 2.0 times and 1.5 times that of cell wall are found to be the most optimal options for R1-HSP and R2-HSP, respectively. To optimize the crashworthiness of the R1-HSP, optimal Latin hypercube design method is used for selection of sampling design points in the design space. And in the meantime, an accurate surrogate modeling method, or more specifically named as response surface method, is adopted to formulate the SEAm, SEAv and σ peak functions. The results yielded from the optimization indicate that the R1-HSP is superior to R0-HSP both in SEAm and SEAv under the same limitation of σ peak. This research is significant in providing technical support in the extended applications of different HSPs used as crashworthiness structures, and it has essential reference value in low volume and low mass energy absorber design.

Negative stiffness honeycombs for recoverable shock isolation

Purpose – The purpose of this paper is to study the behavior of negative stiffness beams when arranged in a honeycomb configuration and to compare the energy absorption capacity of these negative stiffness honeycombs with regular honeycombs of equivalent relative densities. Design/methodology/approach – A negative stiffness honeycomb is fabricated in nylon 11 using selective laser sintering. Its force-displacement behavior is simulated with finite element analysis and experimentally evaluated under quasi-static displacement loading. Similarly, a hexagonal honeycomb of equivalent relative density is also fabricated and tested. The energy absorbed for both specimens is computed from the resulting force-displacement curves. The beam geometry of the negative stiffness honeycomb is optimized for maximum energy absorption per unit mass of material. Findings – Negative stiffness honeycombs exhibit relatively large positive stiffness, followed by a region of plateau stress as the cell walls buckle, similar to regular hexagonal honeycombs, but unlike regular honeycombs, they demonstrate full recovery after compression. Representative specimens are found to absorb about 65 per cent of the energy incident on them. Optimizing the negative stiffness beam geometry can result in energy-absorbing capacities comparable to regular honeycombs of similar relative densities. Research limitations/implications – The honeycombs were subject to quasi-static displacement loading. To study shock isolation under impact loads, force-controlled loading is desirable. However, the energy absorption performance of the negative stiffness honeycombs is expected to improve under force-controlled conditions. Additional experimentation is needed to investigate the rate sensitivity of the force-displacement behavior of the negative stiffness honeycombs, and specimens with various geometries should be investigated. Originality/value – The findings of this study indicate that recoverable energy absorption is possible using negative stiffness honeycombs without sacrificing the high energy-absorbing capacity of regular honeycombs. The honeycombs can find usefulness in a number of unique applications requiring recoverable shock isolation, such as bumpers, helmets and other personal protection devices. A patent application has been filed for the negative stiffness honeycomb design.

Impact resistance and energy absorption of regular and functionally graded hexagonal honeycombs with cell wall material strain hardening

This paper highlights the effects of cell wall material strain hardening and density functional gradation (FG) on in-plane constant-velocity dynamic crushing response and impact behavior of hexagonal honeycombs. Results show that cell wall material strain hardening influences the distinct deformation modes induced by crushing velocity generally observed in regular hexagonal honeycombs. This is seen by a delay in the onset of localized deformation up until intermediate crushing velocities after which localization becomes dominant smearing out differences brought about by cell wall material strain hardening (plasticity convergence). In addition, during the impact loading on regular honeycombs, it was found that increasing the cell wall material strain hardening decreases the rate of gain of maximum crushing strain with increments in initial kinetic energy of impact. On the other hand, introducing FG brings about new deformation patterns due to changes in material distribution and preferential cell wall collapse of the weaker members. Interestingly, although the dynamic localization effect at higher crushing velocities observed earlier was not found to be particularly affected by FG, gradient convergence (i.e. smearing out the effects of FG due to higher velocities analogous to plasticity convergence) was not observed. On the contrary, gradient convergence emerged at higher impacting velocities primarily brought about by a combination of initial deformation localization and its subsequent advancement into FG region ahead. The kinetic energy threshold for the emergence of this gradient convergence effect was found to be considerably delayed by cell wall material strain hardening.