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.     

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.     

Stress analysis of two-dimensional cellular materials with thick cell struts

Finite element analyses (FEA) were performed to thoroughly validate the collapse criteria of cellular materials presented in our previous companion paper. The maximum stress (von-Mises stress) on the cell strut surface and the plastic collapse stress were computed for two-dimensional (2D) cellular materials with thick cell struts. The results from the FEA were compared with those from theoretical criteria of authors. The FEA results were in good agreement with the theoretical results. The results indicate that when bending moment, axial and shear forces are considered, the maximum stress on the strut surface gives significantly different values in the tensile and compressive parts of the cell wall as well as in the two loading directions. Therefore, for the initial yielding of ductile cellular materials and the fracture of brittle cellular materials, in which the maximum stress on the strut surface is evaluated, it is necessary to consider not only the bending moment but also axial and shear forces. In addition, this study shows that for regular cellular materials with the identical strut geometry for all struts, the initial yielding and the plastic collapse under a biaxial state of stress occur not only in the inclined cell struts but also in the vertical struts. These FEA results support the theoretical conclusion of our previous companion paper that the anisotropic 2D cellular material has a truncated yield surface not only on the compressive quadrant but also on the tensile quadrant.     

超弹性蒙皮蜂窝芯的设计及等效模量研究

可变形机翼的蒙皮需要有很高的各向异性,在特定方向具有较低的弹性模量而在其他方向具有较高的气动承载能力,为此设计了一种柔韧性较好、特定方向模量低且承载能力较强的新型超弹性蒙皮结构,对其蜂窝芯单元的力学性能进行了理论计算和有限元分析,并对不同几何参数单个蜂窝芯单元体和多个蜂窝芯单元体进行了轴向刚度测定试验,最后通过理论仿真和试验的对比分析,得到了该超弹性蒙皮蜂窝芯结构轴向刚度的等效模量近似工程计算公式。
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A general constitutive relation for linear elastic foams

A three-dimensional linear elastic constitutive relation is formulated based on a representative unit cell of foam using elasticity theory and micromechanics homogenization scheme. The displacement and strain fields of the unit cell are obtained from elasticity theory and used to derive the macroscopic strain field defined on the outer surface of unit cell through homogenization scheme. By assuming a uniform macroscopic stress on the unit cell surface and the existence of strain energy potential, the constitutive relation of linear elastic foams is obtained. The newly derived constitutive relation is a function of mechanical property of solid constituent, the geometry of cell struts, and the porosity of foams and is able to characterize the anisotropic behavior of foams due to non-uniform strut geometry. The linear elastic response of open-celled foams with both low- and medium-relative densities can be studied using the derived constitutive relation. The effective elastic modulus for uniform strut geometry is reduced from the constitutive relation and agrees well with Gibson and Ashby's semi-empirical equation, Warren and Kraynik's, and Zhu's analytical models within relative density ranging from 0 to 0.35. For non-uniform strut geometry, the calculated effective elastic moduli in three axial directions are different and the material displays anisotropic behavior. The bulk modulus shows less dependence on the variation of the strut geometry. Poisson's ratios are also reduced from the compliance matrix.     

平纹编织C/SiC复合材料的剪切性能

采用IOSIPESCU纯剪切试件,改进反对称四点弯曲装置进行循环加卸载,试验研究了CVI工艺二维平纹编织C/SiC复合材料的面内剪切特性。分析了不同应力水平下卸载模量、残余应变的变化。基于试验研究结果,将剪切应变分解为弹性应变与非弹性应变,分别分析弹性应变和非弹性应变的变化规律,给出了剪切应力应变关系表达式。试件断裂时,最窄截面位置形成平断面,断面扫描电镜分析发现,基体和界面裂纹是主要损伤机理,0°纤维束最后断裂。     

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.     

Elastic lateral-torsional buckling of circular arches subjected to a central concentrated load

An arch under an in-plane central concentrated radial load is subjected to combined axial compressive and bending actions. When these combined axial compressive and bending actions reach a certain value, the arch may suddenly deflect laterally and twist out of its plane of loading and fail in a lateral-torsional buckling mode. This paper derives analytical solutions for the elastic lateral-torsional buckling load of pin-ended circular arches that are subjected to a central concentrated load, using the principle of stationary potential energy in conjunction with the Rayleigh-Ritz method. Analytical solutions of the buckling load for in-plane fixed and out-of-plane pin-ended arches and for the case of the load acting above or below the shear centre are also derived. The analytical solutions are compared with results of a commercial finite element package ANSYS and a finite element code developed by authors elsewhere for arches with different slendernesses, included angles, and cross-sections. The agreement between the analytical solutions and the finite element results is very good.     

Effective elastic constants of two-dimensional cellular materials with deep and thick cell walls

In order to analyse the elastic constants of cellular materials with deep and thick cell walls, finite element analysis using two kinds of unit cell approach (stiffness matrix method and compliance matrix method) is performed which is applicable to any orthotropic cellular materials. Comparison between results from the FEA, the theories presented in this paper and experiments of previous investigators indicate that the elastic constants of cellular materials with thick cell walls depend not only on the relative density but also on the joint stiffening effect. Approximate formulae under generalised plane strain conditions are also presented for the purpose of obtaining the effective elastic constants for cellular materials with deep and thick cell walls. A satisfactory agreement was found with experimental results obtained on a deep and thick cellular material. The results indicate that the previous models in which the wall of cellular materials is treated as a simple beam are not adequate to evaluate the effective elastic constants of cellular materials with deep and thick cell walls. In addition, considerable attention needs for the measurement of effective Young's modulus of square cellular materials in the two soft directions because it is strongly affected by misalignment errors.     

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.     

Basic two-dimensional core types for sandwich structures

The mechanical characteristics of three types of core with two-dimensional isotropic patterns – triangular, hexagonal, and starcell – were studied as related to applications in sandwich structures. The Young's modulus, shear modulus, and Poisson's ratio were calculated for the three core types in the direction normal to the faces. The compressive buckling strength and shear buckling strength were calculated for the three core types by modeling each cell wall of the core as a plate under compressive or shear load. To verify this model, tests were conducted on scaled specimens to measure the compressive buckling strength of each core. The bending flexibilities of the three cores were also studied. Compliances for the three cores were measured using biaxial flexural tests. Tests were performed on each core type in which the deflection of a circular core sample loaded at its center was measured. The three isotropic core patterns exhibited distinct characteristics. In the direction normal to the faces, all three cores had the same stiffness. However, the triangular core had lower compressive and shear buckling strengths than the other two core types. The starcell core exhibited high flexibility compared to the other cores, indicating a potential for application in curved sandwich structures.     

Stiffness of bi-modulus hexagonal and diamond honeycombs

Cellular materials are widely used in various applications because of their low density and high strength. The mechanical behavior of cellular materials under various loading conditions has been investigated. Nevertheless, many of these previous studies assume that the Young’s modulus of constituting struts is the same in tension and compression. The present work first derives analytical expressions for the effective Young’s moduli of hexagonal and diamond lattices composed of struts with different tension and compression moduli under the assumption of small strain deformation. It also uses the finite element method to further investigate the mechanical responses of these lattices. The macroscopic Young’s moduli under both compressive and tensile loads are reported as a function of the ratio of compression and tension moduli of constituting struts. The numerical finite element models are implemented by a user defined material subroutine in ABAQUS. Results reveal that the effective Young’s moduli of periodic hexagonal and diamond lattices significantly decrease with decreasing ratio of compression and tension moduli of the struts. Furthermore, the mechanical behavior of hexagonal lattices composed of struts with different tension-compression moduli is dependent on the loading direction and whether they are compressed or stretched. The unique mechanical properties of bi-modulus cellular materials could find important applications in the automotive and construction industries.

     

基于十四面体模型的开孔泡沫材料弹性模量的有限元分析

利用十四面体模型描述开孔泡沫材料的胞体结构,并用有限元方法确定开孔泡沫材料的弹性模量。计算中使用相同尺寸的十四面体胞体模型,并考虑两种不同支柱截面（圆截面和Plateau截面）形状对弹性模量计算的影响。此外,通过数值方法研究开孔泡沫材料的弹性模量随模型尺寸的变化规律。同时,讨论边界条件处理对开孔泡沫材料弹性模量计算的影响。结果表明,支柱为Plateau截面形状的模型,其弹性模量明显高十具有圆截面支柱模型的结果。且两种模型的弹性模量均随模型尺度的增加而增加,最终趋于一个稳定值,并与Warren和Kraynik的理论预测较为一致。此外,边界条件对模型刚度的影响随着模型尺度的增加而逐渐减小。     

Effects of prebuckling deformations on the elastic flexural-torsional buckling of laterally fixed arches

The effect of the prebuckling in-plane deformations on the elastic flexural-torsional buckling of laterally fixed circular arches is studied in this paper. The finite strains and the energy equation for the flexural-torsional buckling of arches have been derived based on an accurate orthogonal rotation matrix. A closed form solution for the elastic flexural-torsional buckling resistance of laterally fixed arches in uniform bending, including the effects of the prebuckling deformations, is obtained. It is found that the notion that the prebuckling deformations increase the flexural-torsional buckling moment of an arch or of a beam is not necessarily correct for a laterally fixed arch or beam in uniform bending, in deference to a laterally pinned arch. When a laterally fixed arch is subjected to positive uniform bending, the effects of the prebuckling deformations decrease the buckling moment, and the reduction of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When a laterally fixed arch is subjected to negative uniform bending, the effects of the prebuckling deformations decrease the absolute value of its buckling moment when the included angle is very small, but increase the absolute value of the buckling moment when the included angle exceeds a certain value. The increase in the absolute value of the buckling moment increases with an increase of the included angle and of the out-of-plane slenderness ratio of the arch. When the ratio of the out-of-plane to the in-plane second moments of area of the cross-section is not small, both the reduction of the buckling moment of a laterally fixed arch in positive uniform bending and the increase of the buckling moment of a laterally fixed arch in negative uniform bending, are substantial.     

剪切散斑干涉术中剪切量的测量

由于采用剪切散斑干涉术测量离面应变和面内应变都与剪切量有关,所以剪切量的测量精度决定了应变场分布测量的准确性.本文论述了公式法、成像法、莫尔条纹法和相关法等4种剪切量测量法的测量原理,定性分析了它们的测量范围.搭建了一套剪切量测量系统,利用成像法、莫尔条纹法、相关法测量剪切散斑系统的剪切量,并以公式法计算的结果为准,比较其他3种方法的测量结果.结果表明:剪切量＞3 mm时,利用相关法测量具有较高的测量精度;剪切量＜3 mm时,奠尔条纹法测量效果会更好.因此,实际应用中,应该先根据实际测量条件估算剪切量的大致范围,然后依据估算范围选择合适的测量方法.     

Validation of stresses and stress intensity factor in a notched bilayer system under four point bending, as determined by the solution of the Navier's equation

We use a previously formulated solution of the Navier's equation to calculate stress intensity factor in a notched bar bonded to a substrate. We evolve a procedure for determining the stress distribution. We validate the stress analysis in four-point bending by measuring surface strains for different combination of materials in the bilayer (the Young's modulus ratio of the materials in the bilayer varied from 28 to 86). We also determine the stress intensity factor (SIF) experimentally using a notched photoelastic bar bonded to a substrate and validate our estimation of SIF.     

Mechanical properties of fibrous core sandwich panels

Fibrous core sandwich panels are thin, lightweight structures with face sheets separated by an irregular arrangement of independent fibres; the fibres have a random angle of fibre inclination and a range of initial curvatures. These panels have small thickness so that they can be pressed into 3D curvature in a forming operation. This paper analyses the effects of core morphology on the through-thickness elastic moduli, compressive strength and the through-thickness shear strength of the fibrous core. For a specific panel construction, analytical results are compared with both Finite Element analysis and experiment. A new approach to measure the through-thickness shear modulus of fibrous core is described.     

A two-level optimization procedure for material characterization of composites using two symmetric angle-ply beams

A two-level optimization procedure for determining elastic constants E₁, E₂, G₁₂, and ν₁₂ of laminated composite materials using measured axial and lateral strains of two symmetric angle-ply beams with different fiber angles subjected to three-point-bending testing is presented. In the first-level optimization process, the theoretically and experimentally predicted axial and lateral strains of a [(45°/−45°)₆]_s beam are used to construct the strain discrepancy function which is a measure of the sum of the squared differences between the experimental and theoretical predictions of the axial and lateral strains. The identification of the material constants is then formulated as a constrained minimization problem in which the best estimates of shear modulus and Poisson's ratio of the beam are determined to make the strain discrepancy function a global minimum. In the second-level optimization process, shear modulus and Poisson's ratio determined in the first level of optimization are kept constant and Young's moduli of the second angle-ply beam with fiber angles different from 45° are identified by minimizing the strain discrepancy function established at this level of optimization. The suitability of the proposed procedure for material characterization of composite materials has been demonstrated by means of a number of examples.     

Out-of-plane shearing characteristics of coated paperboard

The out-of-plane shearing characteristics of a paperboard by a straight punch and dies are important to estimate the paperboard stripping performance. In this paper, in order to reveal the die clearance effect and the feed velocity effect on the sheared end-profile of the paperboard, a precision shear testing device was developed. The shear cutting of the paperboards was experimentally carried out by varying the die clearance and the feed velocity of punch. Through this work, it was confirmed that (1) the load response had two kinds of transition gradient before passing through the maximum load point; (2) the shearing performance was classified with the normalized indentation depth d/t=0.1∼0.2 into an out-of-plane shear mode and a combined shear mode. Here, the combined mode was composed of out-of-plane bending and in-plane tensile deformation; (3) the surface layers of paperboard were observed with a de-lamimated state using an image processing of side-view flow deformation of the paperboard. Furthermore, (4) the shear strength was discussed with the orthotropic properties of the paperboard.     

Elastic stresses for 90° elbows under in-plane bending

Using finite element analysis, this paper extends elastic stress solutions for 90° pipe elbows under in-plane bending, given in Marie et al. (2007) [1], to cases of mean pipe radius-to-thickness ratio up to 50. It is found that for 90° elbows an in-plane bending moment produces not only an axial membrane stress component but also axial and hoop bending stress components. Furthermore, the magnitudes of these stress components depend strongly on the mean radius-to-thickness ratio, the circumferential location and the longitudinal location. Maximum stresses tend to occur in the centre of the elbow at or near the crown.