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

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

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

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

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

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

Plastic collapse of cellular structures comprised of doubly tapered struts

In this paper, micro-structural models are developed to examine the effects of tapered strut morphology on the plastic collapse of cellular structures. The analytical models are for materials that fail by plastic yielding and cover several types of columnar structure (e.g. hexagonal and square honeycombs, and hexagonal and rhombic cellular materials of rod-like columnar structure). The results indicate that the plastic collapse of hexagonal cellular materials is dependent not only on the relative density but also on the strut morphology. The presence of taper in struts can increase or decrease the plastic collapse strength of cellular materials, depending on the strut morphology.     

Plastic deformation modes of regular hexagonal honeycombs under in-plane biaxial compression

A limit analysis approach is employed to identify the plastic deformation modes of regular hexagonal honeycombs with relatively large wall-thickness-to-length ratios under in-plane biaxial compression. An infinite block of honeycomb material is considered and a representative block consisting of four hexagonal cells is defined when assuming the kinematic admissibility of the modes and a periodic repeatability of the representative block in both spatial directions. In general, three plastic collapse modes are found to be preferable depending on the direction of loading, and in some particular cases they are similar to the modes that occur elastically under stress or strain controlled in-plane biaxial compression. It is shown that the critical forces at the onset of the plastic collapse depend on the assumed constraints for the deformation of the representative block. The results obtained from the theoretical analysis and the numerical simulations are compared and discussed.     

Behavior of intact and damaged honeycombs: a finite element study

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

Effects of solid distribution on the elastic buckling of honeycombs

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

Effective mechanical and transport properties of cellular solids

We utilize two different approaches, homogenization theory and discrete network analyses, to study the mechanical and transport properties of two-dimensional cellular solids (honeycombs) consisting of either hexagonal, triangular, square or Voronoi cells. We exploit results from homogenization theory for porous solids (in the low-density limit) to establish rigorous bounds on the effective thermal conductivity of honeycombs in terms of the elastic moduli and vice versa. It is shown that for hexagonal, triangular or square honeycombs, the cross-property bound relating the bulk modulus to the thermal conductivity turns out to be an exact and optimal result. The same is true for the cross-property bound linking the shear or Young's modulus of the triangular honeycomb to its conductivity. For low-density honeycombs, we observe that all of the elastic moduli do not depend on the Poisson's ratio of the solid phase. The elastic-viscoelastic correspondence principle enables us to conclude that all of the viscoelastic moduli of honeycombs in the low-density limit are proportional to the complex Young's modulus of the solid phase. Such structures have real Poisson's ratios and the loss tangent is the same for any load.     

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.     

Size effects in ductile cellular solids. Part I: modeling

In the mechanical testing of metallic foams, an important issue is the effect of the specimen size, relative to the cell size, on the measured properties. Here we analyze size effects for the modulus and strength of regular, hexagonal honeycombs under uniaxial and shear loadings. Size effects for indentation of a honeycomb are evaluated using finite element analysis. Finally, the results for honeycombs are extrapolated to foams. The results are compared with data for metallic foams in the following, companion paper.     

纸蜂窝压缩密实化应变评估

分析纸蜂窝压缩密实化应变对纸蜂窝吸能特性评估的重要性,引入纸蜂窝压缩瞬时相对密度的概念,从理论上建立纸蜂窝压缩密实化应变评估方程,并做了不同结构参数的纸蜂窝准静态压缩试验,试验结果表明纸蜂窝胞壁厚度、边长和纸蜂窝的拉伸率等结构因素对压缩密实化应变均有一定的影响:纸蜂窝压缩密实化应变随胞壁的厚度δ的增大而减小;随蜂窝胞元边长ι的增大而增大:随蜂窝胞壁厚跨比δ/ι的增大而减小;随其拉伸率γ的增大,先增大后减小,在拉伸率为1时,其压缩密实化应变达最大值;纸蜂窝结构因素对其压缩密实化应变的影响与文中的理论公式是相符的,纸蜂窝的密实化应变与其相对密度近似呈反比例关系,当纸蜂窝的瞬时相对密度为0.39左右时,纸蜂窝压缩趋于密实化.将该评估方程用于五层瓦楞纸板的压缩密实化过程评估表明它也可用于其他结构形式纸蜂窝材料的压缩密实化应变评估,具有一定的普适性.     

OUT-OF-PLANE COMPRESSIVE PROPERTIES OF HEXAGONAL PAPER HONEYCOMBS

The compressive behaviour of paper honeycombs is studied by means of an experimental analysis. Experiment results show how geometry aspects of hexagonal paper honeycombs,e.g. the height of paper honeycomb,the thickness and length of honeycomb cell-wall,the drawing ratio of hexagonal honeycomb,affect the compressive properties of the paper honeycombs. It is in good agreement with the theory model. The constraint factor K of the critical buckling stress is mainly determined by the length of honeycomb cell-wall. It can be described as K=1.54 for B type paper honeycombs and K=3.32 for D type paper honeycombs. The plateau stress is the power exponent function of the thickness to length ratio of honeycomb cell-wall,and the experiment results show that the constant is 13.2 and the power exponent is 1.77. The research results can be used to characterize and improve efficiently the compressive properties of paper honeycombs.     

Elastic collapse of honeycombs under out-of-plane pressure

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

Effect of microstructural topology upon the stiffness and strength of 2D cellular structures

This paper explores the relation between the microstructure and the effective properties of cellular solids. Most available models are based on Voronoi structures, giving a limitation in the cell geometry diversity. In this study, circular cylinder packings followed by radical plane determination leads to various 2D structures exhibiting bimodal or multimodal cell size distributions. These structures are then modelled by a network of beams and are used in a finite element analysis (FEA). Macroscopic properties, such as Young's modulus and the yield strength are estimated. The yield strength corresponds to the appearance of the first plastic hinge. The results of the simulations reveal a large influence of the cell geometry on the mechanical properties. In the case of low densities, scaling laws involving pertinent geometrical characteristics such as beam length or proportion of large cells are proposed to describe Young's modulus and the yield strength.     

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

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

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

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

A morphological elastic model of general hexagonal columnar structures

A general three-dimensional (3D) anisotropic hexagonal model of columnar structure with non-uniform strut morphology is developed. This model covers several types of cellular structure such as two-dimensional (2D) hexagonal and square honeycombs, and 3D hexagonal and rhombic cellular materials of rod-like columnar structure. The effective elastic constants are determined taking account of bending, axial, and shear deformations of the struts. Unlike the theoretical work of other investigators for 2D honeycombs, considering bending, axial and shearing deformations of struts, the present model not only produces transverse isotropy for regular hexagonal columnar structure but also provides a consistent Poisson's ratio when applied to a square honeycomb. The effect of tapered strut morphology on the elastic properties of cellular structures is investigated. For the general hexagonal columnar structures, the bending compliance is the dominant function for the in-plane elastic constants of 2D and 3D structures (excluding the in-plane shear modulus of rhombic structures) and the out-of-plane shear moduli of 3D structures, but the axial compliance is dominant for the in-plane shear modulus of 2D and 3D rhombic structures and the out-of-plane Young's modulus of 3D structures. For cellular materials with the same relative density, the presence of taper increases values of the effective Young's and shear moduli for which the bending compliance is dominant, but decreases those for which the axial compliance is dominant. It is found that the effective elastic properties of cellular materials are dependent not only on the relative density but also on strut morphology both in cross-section geometry and its variation along the strut length which the present model takes account of. These results illustrate the importance of the strut morphology in calculating the effective elastic properties of cellular materials.     

The out-of-plane properties of honeycombs

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

The effect of doubly tapered strut morphology on the plastic yield surface of cellular materials

Although the literature on the mechanics of cellular materials is vast, there is no theoretical model to account for the effects of axial yielding of struts aligned to the applied loading direction on the plastic yield surface under multiaxial loading conditions. An anisotropic hexagonal model having tapered strut morphology is developed to show these effects on the plastic yield surface under multiaxial tensile loading condition. This model covers several types of cellular structure such as two-dimensional (2D) hexagonal and square cellular materials, and three-dimensional (3D) hexagonal and rhombic cellular materials of rod-like columnar structure. A tetrahedral element with tapered strut morphology is also used for a foam model to illustrate these effects on the yield surface under axisymmetric loading condition. Plastic collapse due to bending moment in the inclined struts is a dominant mode. However, under multiaxial tensile loading, the collapse due to axial yielding of struts parallel to the loading direction is found to be an important mode. The shape of plastic yield surface was found to depend not only on relative density but also on the strut morphology.     

Analysis on collapse strength of casing wear

Carrying capacity of the casing will reduce after the casing is worn, which seriously affects the subsequent well drilling, well completion, oil extraction and well repair. A lot of researches on calculation of casing wear collapse strength have been done, but few of them focus on collapsing failure mechanism, and influencing factors and law of collapse strength. So, significant difference between estimated value and actual value of collapse strength comes into being. By theoretical analysis, numerical simulation and actual test, the collapsing failure mechanism of casing wear as well as the influencing factors and laws of collapse strength are investigated, and the investigation results show that collapse of crescent casing wear belongs to "three hinged" instability. The severely-worn position on the casing is yielded into the plastic zone first then deformed greatly, which causes the plastic instability of the whole structure. The casing wear collapse strength presents changes of exponent, power function and linear trend with the residual casing wall thickness, wear radius and axial load, respectively. When the flexibility is less than 10°/30 m, the borehole bending has less impact on casing collapse strength. Thus, the computation model for the casing wear collapsing strength is established by introducing wear radius coefficient and casing equivalent yield strength, at the same time, the model is tested. The test results show that the relative error for the computation model is less than 5%. The research results provide a basis for design of the casing string strength and evaluation of down-hole safety.