A new micromechanical approach of micropolar continuum modeling for 2-D periodic cellular material

In this paper, we present a new united approach to formulate the equivalent micropolar constitutive relation of two-dimensional（2-D） periodic cellular material to capture its non-local properties and to explain the size effects in its structural analysis. The new united approach takes both the displacement compatibility and the equilibrium of forces and moments into consideration, where Taylor series expansion of the displacement and rotation fields and the extended averaging procedure with an explicit enforcement of equilibrium are adopted in the micromechanical analysis of a unit cell.In numerical examples, the effective micropolar constants obtained in this paper and others derived in the literature are used for the equivalent micropolar continuum simulation of cellular solids. The solutions from the equivalent analysis are compared with the discrete simulation solutions of the cellular solids. It is found that the micropolar constants developed in this paper give satisfying results of equivalent analysis for the periodic cellular material.

A Large Deformation Model for the Elastic Moduli of Two-dimensional Cellular Materials

We developed a large deformation model for predicting the elastic moduli of two-dimensional cellular materials. This large deformation rondel was based on the large deflection of the inclined members of the cells of celluar materials, The deflection of the inclined member, the strain of the representative structure and the elastic moduli of two-dimensioned cellular materials were expressed using incomplete elliptic integrals. The experimental results show that these elastic moduli are no longer constant at large deformation, but vary significantly with the strain. A comparison was made between this large deformation model and the small deformation model proposed by Gibson and Ashby.

Closed-form solution for shock wave propagation in density-graded cellular material under impact

Density-graded cellular materials have tremendous potential in structural applications where impact resistance is required.Cellular materials subjected to high impact loading result in a compaction type deformation,usually modeled using continuum-based shock theory.The resulting governing differential equation of the shock model is nonlinear,and the density gradient further complicates the problem.Earlier studies have employed numerical methods to obtain the solution.In this study,an analytical closed-form solution is proposed to predict the response of density-graded cellular materials subjected to a rigid body impact.Solutions for the velocity of the impinging rigid body mass,energy absorption capacity of the cellular material,and the incident stress are obtained for a single shock propagation.The results obtained are in excellent agreement with the existing numerical solutions found in the literature.The proposed analytical solution can be potentially used for parametric studies and for effectively designing graded structures to mitigate impact.

On the stress–strain states of cellular materials under high loading rates

A virtual Taylor impact of cellular materials is analyzed with a wave propagation technique, i.e. the Lagrangian analysis method, of which the main advantage is that no pre-assumed constitutive relationship is required. Time histories of particle velocity, local strain, and stress profiles are calculated to present the local stress-strain history curves, from which the dynamic stress-strain states are obtained. The present results reveal that the dynamic-rigid-plastic hardening （D-R-PH） material model introduced in a previous study of our group is in good agreement with the dynamic stress-strain states under high loading rates obtained by the Lagrangian analysis method. It directly reflects the effectiveness and feasibility of the D-R-PH material model for the cellular materials under high loading rates.

Effects of Cell Microtopology on the In-plane Dynamic Crushing Analysis of Re-entrant Square Cellular Material

With the development of three-dimensional printing technologies, so-called cellular materials have achieved increasingattention due to outstanding properties. Unlike pure solid structures, properties of cellular materials are influenced by bothutilized material and cell microtopology. The present paper proposes a novel type of re-entrant square cellular material.To explore the relationship between microtopology and macrodynamic responses systematically, an explicit dynamic finiteelement simulation method is used. This work starts by constructing theoretical models of relative density employing atwo-dimensional unit cell. Then, the effects of geometric features and configurations on dynamic properties are explored,and simulations indicate that variations of geometric parameters strongly affect properties and that the staggered re-entrantsquares are more stable than the regular re-entrant squares. Subsequently, the effects of the impact velocity on dynamiccrushing behaviors are elaborated. On this basis, the relationship of unit mass energy absorption and geometric features isobtained by employing the response surface method. Furthermore, with targets of maximum unit mass energy absorption andminimum relative density, the optimal structural parameters are achieved by using non-dominated sorting genetic algorithm.The study provides a detailed introduction to dynamic behaviors of cellular materials and guidance to design new structureswith superior characteristics of energy absorption.

Macro-architectured cellular materials： Properties, characteristic modes, and prediction methods

Macro-architectured cellular （MAC） material is defined as a class of engineered materials having configurable cells of relatively large （i.e., visible） size that can be architecturally designed to achieve various desired material properties. Two types of novel MAC materials, negative Poisson＇s ratio material and biomimetic tendon reinforced material, were introduced in this study. To estimate the effective material properties for structural analyses and to optimally design such materials, a set of suitable homogenization methods was developed that provided an effective means for the multiscale modeling of MAC materials. First, a strain-based homogenization method was developed using an approach that separated the strain field into a homogenized strain field and a strain variation field in the local cellular domain superposed on the homogenized strain field. The principle of virtual displacements for the relationship between the strain variation field and the homogenized strain field was then used to condense the strain variation field onto the homogenized strain field. The new method was then extended to a stress-based homogenization process based on the principle of virtual forces and further applied to address the discrete systems represented by the beam or frame structures of the aforementioned MAC materials. The characteristic modes and the stress recovery process used to predict the stress distribution inside the cellular domain and thus determine the material strengths and failures at the local level are also discussed.

Influence of manufacturing process-induced geometrical defects on the energy absorption capacity of polymer lattice structures

Modern additive manufacturing processes enable fabricating architected cellular materials of complex shape,which can be used for different purposes.Among them,lattice structures are increasingly used in applications requiring a compromise among lightness and suited mechanical properties,like improved energy absorption capacity and specific stiffness-to-weight and strength-to-weight ratios.A dedicated modeling strategy to assess the energy absorption capacity of lattice structures under uni-axial compression loading is presented in this work.The numerical model is developed in a non-linear framework accounting for the strain rate effect on the mechanical responses of the lattice structure.Four geometries,i.e.,cubic body centered cell,octet cell,rhombic-dodecahedron and truncated cuboctahedron 2+,are investigated.Specifically,the influence of the relative density of the representative volume element of each geometry,the strain-rate dependency of the bulk material and of the presence of the manufacturing process-induced geometrical imperfections on the energy absorption capacity of the lattice structure is investigated.The main outcome of this study points out the importance of correctly integrating geometrical imperfections into the modeling strategy when shock absorption applications are aimed for.

Structural modeling of sandwich structures with lightweight cellular cores

An effective single layered finite element （FE） computational model is proposed to predict the structural behavior of lightweight sandwich panels having two dimensional （2D） prismatic or three dimensional （3D） truss cores. Three different types of cellular core topology are considered： pyramidal truss core （3D）, Kagome truss core （3D） and corrugated core （2D）, representing three kinds of material anisotropy： orthotropic, monoclinic and general anisotropic. A homogenization technique is developed to obtain the homogenized macroscopic stiffness properties of the cellular core. In comparison with the results obtained by using detailed FE model, the single layered computational model can give acceptable predictions for both the static and dynamic behaviors of orthotropic truss core sandwich panels. However, for non-orthotropic 3D truss cores, the predictions are not so well. For both static and dynamic behaviors of a 2D corrugated core sandwich panel, the predictions derived by the single layered computational model is generally acceptable when the size of the unit cell varies within a certain range, with the predictions for moderately strong or strong corrugated cores more accurate than those for weak cores.

Finite element simulations on the mechanical properties of MHS materials

Finite element simulations are carried out to examine the mechanical behavior of the metallic hollow sphere （MHS） material during their large plastic deformation and to estimate the energy absorbing capacity of these materials under uniaxial compression. A simplified model is proposed from experimental observations to describe the connection between the neighboring spheres, which greatly improves the computation efficiency. The effects of the governing physical and geometrical parameters are evaluated; whilst a special attention is paid to the plateau stress, which is directly related to the energy absorbing capacity. Finally, the empirical functions of the relative material density are proposed for the elastic modulus, yield strength and plateau stress for FCC packing arrangement of hollow spheres, showing a good agreement with the experimental results obtained in our previous study.

Strain-rate effect on initial crush stress of irregular honeycomb under dynamic loading and its deformation mechanism

The seemingly contradictory understandings of the initial crush stress of cellular materials under dynamic loadings exist in the literature, and a comprehensive analysis of this issue is carried out with using direct information of local stress and strain. Local stress/strain calculation methods are applied to determine the initial crush stresses and the strain rates at initial crush from a cell-based finite element model of irregular honeycomb under dynamic loadings. The initial crush stress under constant-velocity compression is identical to the quasi-static one, but less than the one under direct impact, i.e. the initial crush stresses under different dynamic loadings could be very different even though there is no strain-rate effect of matrix material. A power-law relation between the initial crush stress and the strain rate is explored to describe the strain-rate effect on the initial crush stress of irregular honeycomb when the local strain rate exceeds a critical value, below which there is no strain-rate effect of irregular honeycomb. Deformation mechanisms of the initial crush behavior under dynamic loadings are also explored.The deformation modes of the initial crush region in the front of plastic compaction wave are different under different dynamic loadings.

Macrocellular vitreous carbon with the improved mechanical strength

Vitreous carbons with regular macrocellular structure, open intercon- nected porosity, high specific strength and hydraulic permeability were synthesized by infiltration of the epoxy resin into the sacrificial template made from the carbamide granules. Polyvinylpyrrolidone （PVP） solution in ethanol was used as the template binder. When the resin setting and the template extraction had been performed, the resultant porous material was pyrolysed in the nitrogen flow. Depending on PVP concentration in the template binder, final vitreous carbons had the following properties： bulk density at 0.17-0.22g/cm3; porosity at 85.7%-89.0%; window size at 447-735pm; Darcian permeability coefficient at （0.64-9.5）×10-9m2; non-Darcian permeability coefficient at （0.53-3.36）×10-4 m. High specific strength of above 8×10^3 Pa/（kg.m-3） was attained.

Dynamic Responses of Cellular Metal-Filled Steel Beam-Column Joint Under Impact Loading

Taking the excellent energy absorption performances of cellular structures into consideration,three beam-column steel joints are proposed to analyze the effect of cellular metallic fillers on impact mechanical responses of beam-column joints.Based on the existing experimental results,the finite element models of the associated joints are established by using finite element method software.The deformation mode,the bearing capacity and energy absorption performance of various joints subjected to impact loadings with the loading velocities from 10 to 100 m/s are analyzed,respectively.The dynamic responses of cellular metal-filled beamcolumn joints are quantitatively analyzed by means of displacements of central region,nominal impacting stress and energy absorption efficiency.The results can be concluded that the filling of cellular filler weakens the stress concentration on joints,alleviates the occurrence of tearing in connection region among column and beam,and reduces the displacement caused by impact loading.Energy absorption efficiency of filled joints subjected to impact loading increases as the impacting velocity increases,and the cellular metallic filler improves their impact resistance of beam-column joints.The energy absorption efficiency of fully filled joints is superior to that of others.This study can provide a reference for steel structural design and post-disaster repair under extreme working conditions.

Numerical Simulation of the Mechanical Properties of Sintered and Bonded Perforated Hollow Sphere Structures (PHSS)

This paper investigated the uniaxial mechanical properties of a new type of hollow sphere structures. For this new type, the sphere shell was perforated by several holes in order to open the inner sphere volume and surface. The mechanical properties, i.e. elastic properties and initial yield stress of perforated hollow sphere structures （PHSS） in a primitive cubic arrangement were numerically evaluated for different hole diameters and different joining techniques of the hollow spheres. The results are compared to classical configurations without perforation.

A PRECISE METHOD FOR SOLVING WAVE PROPAGATION IN HOLLOW SANDWICH CYLINDERS WITH PRISMATIC CORES

Wave propagation in infinitely long hollow sandwich cylinders with prismatic cores is analyzed by the extended Wittriek-Williams （W-W） algorithm and the precise integration method （PIM）. The effective elastic constants of prismatic cellular materials are obtained by the homogenization method. By applying the variational principle and introducing the dual variables the canonical equations of Hamiltonian system are constructed. Thereafter, the wave propagation problem is converted to an eigenvalue problem. In numerical examples, the effects of the prismatic cellular topology, the relative density, and the boundary conditions on dispersion relations, respectively, are investigated.