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 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.     

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.     

Plastic collapse of thin-walled frusta and egg-box material under shear and normal loading

The quasi-static plastic collapse of thin-walled frusta is determined for combined shear and out-of-plane compression. Experiments and finite element calculations are conducted on conical metallic frusta with semi-included cone angles of 30^ο and 45° to determine the shear collapse response. Additional finite element predictions are given for compressive loading, and for combined shear-compressive loading. The dependence of strength and energy absorption upon geometry is explored. The predicted response of an array of conical frusta is used to give the overall response of an egg-box material sandwiched between rigid face sheets. Scaling laws are determined for the stiffness and strength as a function of relative density of the egg-box material.     

The effect of strut geometry on the yielding behaviour of open-cell foams

A micromechanical analysis was carried out to investigate the effect of strut geometry on the yielding behaviour of open-cell foams. Different strut cross sections, in rectangular, circular and equilateral triangular shapes, were investigated. It was found that the strut geometry significantly affects the plastic-yielding behaviour of open-cell foams. The shape of the plastic-yield surface was found to depend not only on relative density but also on the cross-sectional shape of the struts. Numerical results show that even though the material of the struts is perfectly plastic, open-cell foams with asymmetrical sectional struts will exhibit different tensile and compressive collapse strength.     

Modelling of MHS cellular solid in large strains

The stress–strain characteristics of metal hollow sphere (MHS) material are obtained in relatively large strains under uniaxial compression in two characteristic loading directions. Based on the hypothesis of periodic repeatability of a representative block, large deformations of the material are modelled when assuming point connections between the spheres. The elastic deformations are neglected and a rigid perfectly plastic model is assumed for the base material. A structural approach using the limit analysis and the concept of an equivalent structure are then employed to describe the large plastic deformations during post-collapse process of metal hollow spheres, which undergo mainly a snap-through deformation. Stress vs. material density relationships are proposed for different strain levels in each direction of loading. The obtained results can be used to estimate the energy absorbing capacity of MHS materials under quasi-static loading. The theoretical predictions are compared with some test results and reasonable agreement is shown.     

From local failure toward global collapse of elastic–plastic structures in fluctuating fields

Application of shakedown theory to study the load-bearing capacity of truss structures subjected to varying loads is presented. Inadaptation may cause local fractures not leading to the global collapse (the loss of load-bearing capacity) of a structure. A full analysis requires a step-by-step application of the reduced kinematic formulae constructed recently by the author to check the occurrence of a local fracture by alternating plasticity and a possible spreading of the fracture zone until the critical state of global incremental (or instantaneous) collapse is reached. This basic phenomenon, in somehow more sophisticated appearance, might be observed in many more general structures and in inhomogeneous materials working in changing fields, as in some fiber bundle models presented. The solution procedure could also help to improve the design of a structure for particular working conditions.