The effects of hierarchy on the in-plane elastic properties of honeycombs

Introducing hierarchy into structures has been credited with improving elastic properties and damage tolerance. Specifically, adding hierarchical sub-structures to honeycombs, which themselves have good-density specific elastic and energy-absorbing properties, has been proposed in the literature. An investigation of the elastic properties and structural hierarchy in honeycombs was undertaken, exploring the effects of adding hierarchy into a range of honeycombs, with hexagonal, triangular or square geometry super and sub-structure cells, via simulation using finite elements. Key parameters describing these geometries included the relative lengths of the sub- and super-structures, the fraction of mass shared between the sub- and super-structures, the co-ordination number of the honeycomb cells, the form and extent of functional grading, and the Poisson’s ratio of the sub-structure. The introduction of a hierarchical sub-structure into a honeycomb, in most cases, has a deleterious effect upon the in-plane density specific elastic modulus, typically a reduction of 40 to 50% vs a conventional non-hierarchical version. More complex sub-structures, e.g. graded density, can recover values of density specific elastic modulus. With careful design of functionally graded unit cells it is possible to exceed, by up to 75%, the density specific modulus of conventional versions. A negative Poisson’s ratio sub-structure also engenders substantial increases to the density modulus versus conventional honeycombs.     

Elastic properties of chiral,anti-chiral,and hierarchical honeycombs:A simple energy-based approach

The effects of two geometric refinement strategies widespread in natural structures, chirality and self-similar hierarchy, on the in-plane elastic response of two-dimensional honeycombs were studied systematically. Simple closed-form expressions were derived for the elastic moduli of several chiral, antichiral, and hierarchical honeycombs with hexagon and square based networks. Finite element analysis was employed to validate the analytical estimates of the elastic moduli. The results were also compared with the numerical and experimental data available in the literature. We found that introducing a hierarchical refinement increases the Young's modulus of hexagon based honeycombs while decreases their shear modulus. For square based honeycombs, hierarchy increases the shear modulus while decreasing their Young's modulus. Introducing chirality was shown to always decrease the Young's modulus and Poisson's ratio of the structure. However, chirality remains the only route to auxeticity. In particular, we found that anti-tetra-chiral structures were capable of simultaneously exhibiting anisotropy, auxeticity,and remarkably low shear modulus as the magnitude of the chirality of the unit cell increases.     

Experimental investigation of chimney-enhanced natural convection in hexagonal honeycombs

The natural convective heat transfer performance of an aluminum hexagonal honeycomb acting as a novel heat sink for LED cooling is experimentally investigated. The concept of adding an adiabatic square chimney extension for heat transfer enhancement is proposed, and the effects of chimney shape, height, and diameter are quantified. The average N u_av of a heated honeycomb with straight chimney is significantly higher than that without chimney, and the enhancement increases with increasing chimney height. At a given chimney height, honeycombs with divergent chimneys perform better than those with convergent ones. For a fixed divergent angle, the N u_av number increases monotonically with increasing chimney height. In contrast, with the convergent angle fixed, there exists an optimal chimney height to achieve maximum heat transfer.     

A nonlocal continuum model for the buckling of carbon honeycombs

Using molecular dynamics (MD) simulations and Eringen’s nonlocal elasticity theory, in this paper we comprehensively study the small-scale effects on the buckling behaviours of carbon honeycombs (CHCs). The MD simulation results show that the small-scale effects stemming from the long-range van der Waals interaction between carbon atoms can significantly affect the buckling behaviours of CHCs. To incorporate the small-scale effects into the theoretical analysis of the buckling of CHCs, we develop a nonlocal continuum mechanics (CM) model by employing Eringen’s nonlocal elasticity theory. Our nonlocal CM model is found to fit MD simulations well by setting the nonlocal parameter in the nonlocal CM model as 0.67. It is shown in our MD-based nonlocal CM model that when the cell length of CHCs is smaller than 7.93 Å the influence of small-scale effects on the bucking of CHCs becomes unnegligible and the small-scale effects can greatly reduce the critical buckling stress of CHCs. This reduction in critical buckling stress induced by the small-scale effects becomes more significant as the length of the cell wall decreases. Moreover, CHCs are found to display two different buckling modes when they are under different states of loading. The critical condition for the transition between these two buckling modes of CHCs can be greatly affected by the small-scale effects when the vertical cell wall and the inclined cell wall of CHCs have different lengths.     

Optimal control of free-stream turbulence intensity by means of honeycombs

The characteristics of honeycombs of different shapes and sizes used for reducing free-stream turbulence in wind tunnels have been experimentally investigated. The optimum geometric dimensions of the honeycomb and its optimum location in the wind tunnel necessary to ensure a minimum level of turbulence in the working section have been determined.Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 163–174, May–June, 1994.     

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.     

Inertia effects on the progressive crushing of aluminium honeycombs under impact loading

This paper presents the test results under quasi-static and impact loadings for a series of aluminum honeycombs (3003 and 5052 alloys) of different cell sizes, showing significantly different enhancements of the crushing pressure between 3003 honeycombs and the 5052 ones. A comprehensive numerical investigation with rate insensitive constitutive laws is also performed to model the experimental results for different cell size/wall thickness/base material, which suggests that honeycomb crushing pressure enhancement under impact loading is mostly due to a structural effect.Such simulated tests provide detailed local information such as stress and strain fields (in the cell wall) during the whole crushing process of honeycombs. A larger strain (in the cell wall) under impact loading than for the quasi-static case before each successive folding of honeycombs is observed, because of the lateral inertia effect. Thus, differences of the ratios of the stress increase due to strain hardening over the yield stress between 3003 and 5052 alloys lead to the different enhancements of crushing pressure. This result illustrates that the lateral inertia effect in the successive folding of honeycombs is the main factor responsible for the enhancement of the crushing pressure under impact loading.     

Impact behavior of honeycombs under combined shear-compression. Part I: Experiments

This paper presents a combined shear-compression impact test for soft cellular materials designed in order to investigate their behavior under impact multiaxial loadings. A large-diameter Nylon Split Hopkinson Pressure Bar system (SHPB) with beveled ends of different angles is used to apply the desired shear-compression combinations. The data processing methods are studied and validated by virtual testing data generated with FEM simulations. A series of experiments on an aluminum honeycomb were performed at the impact velocity of about 15 m/s with the loading angles ranging from 0° (corresponding to the pure compression) to 60°. It shows a strong effect of the additional shear loading because both the initial peak and the crush strength decrease with increasing loading angles. The quasi-static shear-compression experiments were also performed using the same beveled ends on a universal INSTRON machine and a notable strength enhancement under impact loading is observed. Images captured during quasi-static and impact tests permit for the determination of the two co-existing deforming patterns under combined shear-compression and reveal the influence of the loading rate on the occurrence of these two patterns.     

Impact behavior of honeycombs under combined shear-compression. Part II: Analysis

In this paper, a numerical virtual model of honeycomb specimen as a small structure is used to simulate its combined shear-compression behavior under impact loading. With ABAQUS/Explicit code, the response of such a structure made of shell elements is calculated under prescribed velocities as those measured in the combined shear-compression tests presented in Part I of this study.The simulated results agree well with the experimental ones in terms of overall pressure/crush curves and deformation modes. It allows for the determination of the separated normal behavior and shear behavior of honeycomb specimen under dynamic combined shear-compression. It is found that the normal strength of honeycombs decreases with increasing shearing load. Quasi-static calculations were also performed and a significant dynamic strength enhancement found in experiments was validated again in the numerical work. A crushing envelope in normal strength vs. shear strength plane was obtained on the basis of these simulations.     

Analytical design of effective thermal conductivity for fluid-saturated prismatic cellular metal honeycombs

A comparative optimal design of fluid-saturated prismatic cellular metal honeycombs(PCMHs) having different cell shapes is presented for thermal management applications. Based on the periodic topology of each PCMH, a unit cell(UC) for thermal transport analysis was selected to calculate its effective thermal conductivity. Without introducing any empirical coefficient, we modified and extended the analytical model of parallel–series thermal–electric network to a wider porosity range(0.7 ~ 0.98) by considering the effects of two-dimensional local heat conduction in solid ligaments inside each UC. Good agreement was achieved between analytical predictions and numerical simulations based on the method of finite volume. The concept of ligament heat conduction efficiency(LTCE) was proposed to physically explain the mechanisms underlying the effects of ligament configuration on effective thermal conductivity(ETC).Based upon the proposed theory, a construct strategy was developed for designing the ETC by altering the equivalent interaction angle with the direction of heat flow: relatively small average interaction angle for thermal conduction and relatively large one for thermal insulation.     

Hierarchical micro-nanostructures from polyoxometalates and polydopamine:Characterization,electrochemical and intrinsic peroxidase-like properties

Hybrid composite materials with hierarchical structures have attracted continuing attention for their enhanced features and various applications.The new hybrid ...     

The effect of cell irregularity on the high strain compression of 2D Voronoi honeycombs

The in-plane compression of low-density irregular Voronoi honeycombs with periodic boundary conditions has been simulated to engineering strains of 0.6 using finite element analysis. Different degrees of geometric irregularity in the honeycomb cells, as quantified using a regularity parameter, have been employed. The stress–strain predictions reveal that, for a fixed relative density, a more irregular honeycomb has a higher tangential modulus at low strain but supports a lower compressive stress at higher strain (above approximately 0.04) when compared with a more regular honeycomb. A combined ‘springs in parallel’ and ‘springs in series’ model has also been compared quantitatively with the simulation stress–strain results, the relative importance of the ‘springs in series’ mechanism having been found to increase with the irregularity of the honeycomb and, in many cases, with the applied compressive strain. In addition, the dependency of the Poisson’s ratio, the maximum bending strain in the cell walls, and the mean junction rotation upon the applied compressive strain have also been determined for a range of honeycomb irregularities.     

Hierarchical modeling in multibody dynamics

Summary In this paper a hierarchical approach using several mechanical models with different complexities and modeling depths to describe a single engineering system is presented. The mechanical models are derived from (but not limited to) multibody dynamics. The computer power available and improvements in theoretical understanding allow today not only to perform analyses but also to attack the problem of multimodel synthesis. Therefore, hierarchical modeling is used as a basis to analyze simultaneously models with different complexities and different excitations, and to optimize the performance with the most appropriate model for an investigated mechanical effect. Since only one single engineering system is investigated, its different models must be coupled by shared parameters, and the different criteria have to be combined with multicriteria optimization algorithms in order to obtain a single feasible design. An example taken from vehicle dynamics demonstrates the application of the approach. Received 14 January 1997; accepted for publication 11 September 1997     

Role of cell regularity and relative density on elasto-plastic compression response of random honeycombs generated using Voronoi diagrams

The Voronoi tessellation technique and solid modeling methods are used in this work to create virtual random structures and link cell morphology with the mechanical behavior. Their compression responses are analyzed using the finite element method. First, the effect of loading direction is analyzed for structures with different levels of randomness characterized by a regularity parameter to assess the degree of scatter in the results. Subsequently, morphological characteristics such as arrangement of cells and randomness are analyzed separately. The effect of relative density on structures with different levels of randomness is also studied. Simulations suggest that at low relative densities the arrangement of cells has a negligible effect on the compression response of random honeycombs. On the contrary, the cellular randomness has significant influence on the elastic and plastic characteristics especially when fully random structures are compared with the regular counterparts.     

Two-dimensional Hierarchical Models for Prismatic Shells with Thickness Vanishing at the Boundary

We construct variational hierarchical two-dimensional models for elastic, prismatic shells of variable thickness vanishing at boundary. With the help of variational methods, existence and uniqueness theorems for the corresponding two-dimensional boundary value problems are proved in appropriate weighted functional spaces. By means of the solutions of these two-dimensional boundary value problems, a sequence of approximate solutions in the corresponding three-dimensional region is constructed. We establish that this sequence converges in the Sobolev space H¹ to the solution of the original three-dimensional boundary value problem. Mathematics Subject Classifications (2000) 74K20, 74K25.     

Hierarchical structures in a turbulent pipe flow

A hierarchical structure (HS) analysis (β-test and γ-test) is applied to a fully developed turbulent pipe flow. Velocity signals are measured at two cross sections in the pipe and at a series of radial locations from the pipe wall. Particular attention is paid to the variation of turbulent statistics at wall units 10<y⁺<3000. It is shown that at all locations the velocity fluctuations satisfy the She–Leveque hierarchical symmetry (Phys. Rev. Lett. 72 (1994) 336). The measured HS parameters, β and γ, are interpreted in terms of the variation of fluid structures. Intense anisotropic fluid structures generated near the wall appear to be more singular than the most intermittent structures in isotropic turbulence and appear to be more outstanding compared to the background fluctuations; this yields a more intermittent velocity signal with smaller γ and β. As turbulence migrates into the logarithmic region, small-scale motions are generated by an energy cascade and large-scale organized structures emerge which are also less singular than the most intermittent structures of isotropic turbulence. At the center, turbulence is nearly isotropic, and β and γ are close to the 1994 She–Leveque predictions. A transition is observed from the logarithmic region to the center in which γ drops and the large-scale organized structures break down. We speculate that it is due to the growing eddy viscosity effects of widely spread turbulent fluctuations in a similar way as in the breakdown of the Taylor vortices in a turbulent Couette–Taylor flow at high Reynolds numbers.     

Identifying Multidimensional Damage in a Hierarchical Dynamical System

In this paper, we present a novel method for multidimensional damage identification based on a dynamical systems approach to damage evolution. This approach does not depend on the knowledge of particular damage physics, and is appropriate for systems where damage evolves on much slower time scale than the directly observable dynamics. In an experimental context, the phase space reconstruction and locally linear models are used to quantify small distortions occurring in a dynamical system's phase space due to damage accumulation. These measurements are then related to the drifts in damage variables. A mathematical model of a harmonically driven cantilever beam in a force field of two battery-powered electromagnets is used to demonstrate validity of the method. It is explicitly demonstrated that an affine projection of the described feature vector accurately tracks the two competing damage processes. For practical damage identification purposes, the tracking data is analyzed using the proper orthogonal decomposition (POD) and smooth orthogonal decomposition (SOD) methods. Both methods correctly identify the two dominant damage modes. However, the SOD is more impervious to changes in fast-time dynamics and provides a significantly better signal-to-noise ratio. The damage modes identified using SOD are demonstrated to be within a linear transformation from the actual damage states and can be used to reconstruct the corresponding phase space trajectory.