Voronoi Honeycombs Research Articles

Density-graded Voronoi honeycombs – A local transversely isotropic description

Density-graded cellular materials, in which their relative density varies with location, are amenable to design for multi-functions, and have been used in mechanical buffers and protective structures. To estimate the force–deformation responses of density-graded cellular materials efficiently without resorting to modelling individual cells, a constitutive model based on local transverse isotropy is adopted. Metallic Voronoi honeycomb-like specimens with cell sizes that vary with position, are fabricated via 3D printing, to represent a density-graded cellular material with local transverse isotropy that varies with position. From quasi-static compression tests and cell-based finite element (FE) simulations, the influence of relative density on the continuum constitutive model is investigated and demonstrated to be significant. The model is then used to predict the global stress–strain response and deformation of cellular materials with positive and negative density-gradients, defined along the loading direction, subjected to uniaxial quasi-static compression, quasi-static indentation, and dynamic compression. The results show that for quasi-static compression, severe plastic deformation progresses from the low relative density region to the high relative density region; however, impact loading and indentation of negative density-gradient Voronoi honeycombs generates two zones of gross deformation – in the vicinity of the applied load and the opposite lower density end. Predictions based on the proposed constitutive model display good correlation with experimental and cell-based FE results.

Mechanical and energy absorption properties of 3D-printed honeycomb structures with Voronoi tessellations

3D printing technology is the new frontier in building construction. It is especially useful for making small structures within a short period. Full construction, including interior partitions and exterior façades, can be achieved with this technology. This paper proposes a parametric Voronoi tessellations model for quickly generating and fabricating 3D-printed hexagonal honeycomb partitions for interior design. Comprehensive experimental testing was conducted to characterize the mechanical properties and investigate the energy absorption characteristics of the proposed 3D-printed hexagonal honeycomb while comparing it to alternative hexagonal honeycomb structures. The tests included tensile testing (ASTM-D638) of the printed Polylactic Acid (PLA) material, especially with the almost total absence of conducted research that reported mechanical properties for 3D printed material with low infill percentages such as 10%. In addition, an in-plane quasi-static axial compression testing of the lightweight honeycomb structures was also conducted on the printed structure with the same low infill percentage. Compared to non-Voronoi honeycomb structures, the Voronoi honeycomb resulted in superior mechanical and energy absorption properties with energy absorption values ranging from 350 to 435 J and crash force efficiency being 1.42 to 1.65.

A note on the modelling of foams using Voronoi technique

Previous researchers have used Voronoi honeycombs to represent and predict the responses of random foams. Often, a two-dimensional Voronoi honeycomb model is used and the results are regarded as being identical to those of actual foams, which are in fact three-dimensional. The present study focuses on possible performance difference and the intrinsic relationship between Voronoi honeycombs and foams. Geometrical models of Voronoi honeycombs and foams with different cell irregularities are generated. Their relative densities are first calculated and compared. Based on the geometric models and an idealised elastic perfectly plastic property for the parent material, finite element (FE) analysis is carried out, respectively, for the block of the cellular material under quasi-static compression between two rigid platens. Consequently, stress–strain relationships are obtained, and it is shown that a two-dimensional Voronoi model cannot always be adopted to represent a three-dimensional foam in studying the behaviour of cellular solids.

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.

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.

Stress concentration around an atelectatic region: A finite element model

Lung parenchyma surrounding an atelectatic region is thought to be subjected to increased stress compared with the rest of the lung. Using 37 hexagonal cells made of linear springs, Mead et al. (1970) measured a stress concentration greater than 30% in the springs surrounding a stiffer central cell. We re-examine the problem using a 2D finite element model of 500 cells made of thin filaments with a non-linear stress–strain relationship. We study the consequences of increasing the central stiff region from one to nine contiguous cells in regular hexagonal honeycombs and random Voronoi honeycombs. The honeycomb structures were uniformly expanded with strains of 15%, 30%, 45% and 55% above their resting, non-deformed geometry. The curve of biaxial stress vs. fractional area change has a similar shape to that of the pressure–volume curve of the lung, showing an initial regime with relatively flat slope and a final regime with decreasing slope, tending toward an asymptote. Regular honeycombs had little variability in the maximum stress in radially oriented filaments adjacent to the central stiff region. In contrast, some filaments in random Voronoi honeycombs were subjected to stress concentration approximately 16 times the average stress concentration in the radially oriented filaments adjacent to the stiff region. These results may have implications in selecting the appropriate strategy for mechanical ventilation in ARDS and defining a “safe” level of alveolar pressure for ventilating atelectatic lungs.

Dynamic Crushing of Voronoi Honeycombs: Local Stress-Strain States

Dynamic stress-strain states in Voronoi honeycombs are investigated by using cell-based finite element models. Two different loading scenarios are considered: the high-constant-velocity compression and the direct impact. The 2D local engineering strain fields are calculated. According to the feature of shock front propagation, the 1D distribution of local engineering strain in the loading direction is deduced from the 2D strain fields, which provide evidences of the existence of discontinuities at shock front in cellular materials and thus enhance the basis of the continuum-based shock models. A method to quantitatively clarify the local stress-strain states ahead of and behind the shock front is developed. The results show that the dynamic stress-strain states in the densification stage obtained from both loading scenarios are different from the quasi-static stress-strain relation. The stress ahead of the shock front obtained from the high-constant-velocity compression scenario is slightly smaller than the quasi-static yield stress, but that obtained from the direct impact scenario is larger than the quasi-static yield stress. The possible mechanisms of deformation and wave propagation are explored.

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.

Dynamic compressive strength properties of aluminium foams. Part II—‘shock’ theory and comparison with experimental data and numerical models

One-dimensional ‘steady-shock’ models based on a rate-independent, rigid, perfectly-plastic, locking ( r- p- p- l) idealisation of the quasi-static stress–strain curves for aluminium foams are proposed for two different impact scenarios to provide a first-order understanding of the dynamic compaction process. A thermo-mechanical approach is used in the formulation of their governing equations. Predictions by the models are compared with experimental data presented in the companion paper (Part I) and with the results of finite-element simulations of two-dimensional Voronoi honeycombs. A kinematic existence condition for continuing ‘shock’ propagation in aluminium foams is established using thermodynamics arguments and its predictions compare well with the experimental data. The thermodynamics highlight the incorrect application of the global energy balance approach to describe ‘shock’ propagation in cellular solids which appears in some current literature.

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.

Modeling the Contact Stiffness Between a 2D Voronoi Honeycomb and a Flat Rigid Surface

ABSTRACTOpen-cell foam can be thought of as a network of interconnected struts. To study the contact stiffness behavior, one can imagine the foam boundary to be characterized by struts that are free at one end, while the base-end is connected to the interior of the foam. In previous studies, the base-end was assumed to be built into a rigid surface, i.e., the surface struts were constrained to have zero displacement where they would connect to the foam interior. In this study the assumption that the surface elements are built-in is relaxed by modeling the elastic behavior of the foam interior. The foam interior is modeled as a network of struts within a finite element formulation. The analysis is simplified by considering two-dimensional honeycomb structures. It is found that including the foam interior in the model results in a significant reduction in the predicted contact stiffness.

Effects of cell irregularity on the elastic properties of 2D Voronoi honeycombs

Foams are more and more widely used in different areas. Most available mechanical models are usually based on idealised unit cell structures, and are not able to account for the natural variations in microstructure which are typical for most foam structures. The objective of this work has been to investigate how the cell irregularity affects the elastic properties of 2D random foams. We have constructed periodical random structures with different degrees of irregularity, and applied finite element analysis (FEA) to determine their effective elastic properties. The results indicate that, the more irregular the 2D random foams, the larger will be their effective Young's modulus and shear modulus, and the smaller will be their bulk modulus at a constant overall relative density. However, for varying degrees of irregularity, the foams remain isotropic and the Poisson's ratios are very close to 1. Both Young's modulus and Poisson's ratio of random Voronoi honeycombs having different degrees of regularity α, decrease gradually with increasing relative density ρ.

The role of cellular structure in creep of two-dimensional cellular solids

The creep response of cellular solids is sensitive to the details of the microstructure. Here, finite element simulations were used to model the steady state, secondary creep rate of several two-dimensional cellular solids: a Voronoi honeycomb, representing a structure with a random variation in cell shape; a plane section of a micrograph of a commercially available closed-cell aluminum foam; and the same plane section of the foam, but with the curvature of the cell walls suppressed. The solid was assumed to follow power-law creep. Both periodic boundary conditions and boundary conditions corresponding to a finite size sample were analyzed. The results of the models are compared with the analytical results for a regular hexagonal honeycomb of the same relative density. The creep rates of all of the other structures are higher than that of the regular hexagonal honeycomb. The results indicate that the details of the microstructure can have a significant effect on the creep rate, and thus the lifetime, of the cellular solid. Cell wall curvature plays the most significant role, but the distribution of cell shape and size also influences the creep rate.

Modelling fatigue damage accumulation in two-dimensional Voronoi honeycombs

Trabecular bone, a porous, cellular type of bone found at the ends of the long bones and within the vertebrae, is subject to cyclic compressive loading resulting from the activities of daily living. Such fatigue loading can result in fracture, especially in vertebrae of patients with osteoporosis. As an initial step in understanding compressive fatigue of trabecular bone we previously used finite-element analysis to model the progressive damage and failure of a simple, two-dimensional hexagonal honeycomb. In this study, the analysis is extended to a random, Voronoi honeycomb. Bending of the cell walls induces tensile stresses even when the overall loading is compressive. The cell walls are assumed to have a distribution of crack lengths in their tensile zones. The cracks are assumed to grow according to a Paris law and fail when the cracks reach 75% of the cell wall thickness. Failed cell walls are removed from the structure, the stress distribution recalculated and the next increment of fatigue loading are simulated. The Young's modulus of the honeycomb is calculated after each cell wall failure. Overall failure of the Voronoi structure is assumed to occur when the modulus is reduced by 5%; further loading reduces the modulus sharply. The slope of the S–N curve for the Voronoi honeycomb is the same as that for the hexagonal honeycomb. The model suggests that a random honeycomb is more sensitive to fatigue than a regular one.

The effects of non-periodic microstructure and defects on the compressive strength of two-dimensional cellular solids

The utility of unit cell models that assume periodic microstructures may be limited when applied to cellular materials that have non-periodic microstructures. We analyzed the effects of non-periodic microstructure and defects on the compressive failure behavior of Voronoi honeycombs using finite element analysis. Our results indicate that the non-periodic arrangement of cell walls in random Voronoi honeycombs (with cells approximately uniform in size) results in higher strains in a small number of cell walls compared to periodic, hexagonal honeycombs. Consequently, the Voronoi honeycombs were approximately 30% weaker than periodic, hexagonal honeycombs of the same density. The strength difference between the Voronoi and periodic honeycombs depended slightly on density, due to density-dependent interactions between failure modes (i.e. plastic collapse and elastic buckling). Defects, introduced by removing cell walls at random locations, caused a sharp decrease in the effective mechanical properties of both Voronoi and periodic honeycombs (e.g. a 10% reduction in density due to defects caused a 60% reduction in the strength of Voronoi honeycombs). The sensitivity to defects was comparable for thin-walled, elastomeric honeycombs (relative density 0.015) and for thicker walled, plastic honeycombs (relative density 0.15). The properties degraded to zero when 35% of the cell walls were removed, consistent with the percolation limit for a two-dimensional network of hexagonal cells. When four or more adjacent cell walls were removed, the localized band of cell collapse passed through the defect site and the effective strength and modulus were reduced, indicating that even those defects which have a negligible effect on density can alter the failure pattern as well as the effective properties of honeycombs with cells of approximately equal size and strength.

The effects of non-periodic microstructure on the elastic properties of two-dimensional cellular solids

Models of two-dimensional cellular solids are based on idealized unit cells intended to represent the microstructural features of an average cell from a real material. A significant limitation of the unit cell modelling approach is that it does not account for the natural variations in microstructure which are typical of most cellular materials. Our objective was to model one type of microstructural variability—the non-periodic arrangement of cell walls—and to investigate how this variability affected the relations between microstructure and elastic properties in two-dimensional cellular materials (honeycombs). Specifically, we asked: (1) How much variance in the elastic properties does variability in the arrangement of cells walls introduce? (2) Are the relations between microstructural and elastic properties for non-periodic honeycombs the same, on average, as those for periodic honeycombs? (3) Can anisotropy of elastic properties for non-periodic honeycombs be directly related to microstructural anisotropy, as characterized by stereological parameters? We generated models of non-periodic arrays of Voronoi cells with uniform cell wall thickness, and performed finite element analysis (FEA) to determine effective elastic moduli for low density honeycombs (relative densities < 0.3). Our results indicate that variability in the arrangement of cell walls introduces a modest amount of variance in the elastic constants of isotropic, Voronoi honeycombs (coefficients of variation 4–9%). One consequence of this variance is that differences in the elastic constants in the two directions of nominally isotropic honeycombs are comparable to differences between different honeycombs. Nevertheless, the structure-property relations for both isotropic and anisotropic non-periodic honeycombs are, on average, not different than those for periodic honeycombs. We also measured relative density and microstructural anisotropy using stereology, and demonstrated that the elastic properties of non-periodic honeycombs can be well predicted using this independent estimate of microstructural properties combined with simple relations developed for periodic honeycombs. These findings support the use of the unit cell modelling approach for the analysis of honeycombs of low relative density with random variations in the arrangement of cell walls.

THE STRUCTURE OF AGGREGATES AND THE MOLECULAR KINEMATICS OF THE VISCOSITY OF A BERNAL LIQUID

The structure of 3-dimensional aggregates is discussed as a set of points on which graphs are constructed. By constructing the Voronoi honeycomb (Dirichlet regions) for the points and applying a small "irregularizing transformation", a "simplicial graph" and a "primitive coordination number" (whose value is close to 14 for all aggregates) can be defined universally for both regular and irregular aggregates. Recent studies of the geometry of irregular aggregates (of steel balls, crystal grains, etc.) are reviewed. The theory of liquids of J. D. Bernal is discussed and the simplicial graph is used to show that the "activation volume" of a Bernal liquid is about one-tenth of the molecular volume. The kinematics of flow of aggregates is discussed in terms of their graphs and in terms of a process of "volume exchange"—the production and destruction of free volume. Using these concepts, an equation is derived for the viscosity of a Bernal liquid as a product of five terms expressing respectively the kinematic, stoichiometric, kinetic, pressure-dependent, and shear-dependent factors.