Shear Modulus Determination of Cuboid Metallic Open‐Lattice Cellular Structures by Analytical,Numerical and Homogenisation Methods

Abstract: The shear response of irregular open‐lattice cellular cores made of interconnected metallic struts is analysed, and the core shear modulus in the three material principal directions is derived. The analytical approach is based on the technical beam theory, which is used for the determination of the unit‐cell response under complex loading. The influence of the strut geometrical parameters and unit‐cell shape and size on the cellular core shear stiffness is determined. The analytical determination of the unit‐cell properties is successfully validated by a reference numerical model of the unit cell, which is developed for comparison purposes. Furthermore, the homogenisation principles are applied to the prediction of the shear response of a core block structure comprising a high number of unit cells, for which experimental results were available; the comparison revealed that the experimental results coincide well with the results obtained from the homogenised model.     

Global‐local fatigue assessment of an ancient riveted metallic bridge based on submodelling of the critical detail

Increasing traffic demands (ie, load intensity and operational life) on ancient riveted metallic bridges and the fact that these bridges were not explicitly designed against fatigue make the fatigue performance assessment and fatigue life prediction of riveted bridges a concern. This paper proposes a global‐local fatigue analysis method that integrates beam‐to‐solid submodeling, elastoplastic of material in local region, and local fatigue life prediction approach. The proposed beam‐to‐solid submodeling can recognize accuracy local stress/strain information accompanying with the global structural effect on the fatigue response of local riveted joints. The fatigue life is predicted based on cumulative damage rule, local strains, and number of cycles with consideration of traffic data, where the relation between the fatigue life and local strain is derived according to the Basquin and Manson‐Coffin law. Besides, the elastoplastic of material is considered. The proposed methodology for fatigue life prediction based on local strain parameter and the Palmgren‐Miner linear damage hypothesis is implemented in a case study of an ancient riveted bridge.     

单层网壳结构弹塑性屈曲分析的离散单元法研究

该文提出离散元塑性区法,即将任意2个球元的接触截面划分成若干小面积,通过各小面积的应力状态描述整个截面的塑性发展过程,较离散元塑性铰法更精确。该文推导了杆系离散元截面应变增量计算公式,建立了截面在三维应力-应变状态下的结构弹塑性本构方程、加卸载准则、截面内力积分公式以及计算分析流程。离散元弹塑性屈曲分析的追踪策略与弹性屈曲分析完全相同,即仍采用离散元力控制法或位移控制法。采用Fortran语言自编程序对若干单层网壳结构算例进行弹塑性屈曲分析,验证了离散元塑性区法的正确性和适用性,拓宽了离散单元法在工程领域的应用范围,为结构分析提供了新路径。     

Probabilistic evaluation of fuselage-type composite structures

A methodology is developed to simulate computationally the uncertain behavior of composite structures. The uncertain behavior includes buckling loads, natural frequencies, displacements, stress/strain, etc., which are the consequences of the random variation (scatter) of the primitive (independent random) variables in the constituent, ply, laminate and structural levels. This methodology is implemented in a computer code IPACS (integrated probabilistic assessment of composite structures). A fuselage-type composite structure is analyzed to demonstrate the code's capability. The probability distribution functions of the buckling loads, natural frequency, displacement, strain and stress are computed. The sensitivity of each primitive (independent random) variable to a given structural response is also identified from the analyses.     

On the Characterisation of Foam and Micro‐lattice Materials used in Sandwich Construction1

Abstract: The paper gives an overview of issues relating to the characterisation of the progressive collapse of core cellular materials used in sandwich construction. The specific structural application addressed is foreign object impact, and in this case the core cellular material is subject to multi‐axial stresses, progressive collapse and possible rupture. The paper gives an overview of various theoretical and modelling issues, which are then related to experimental materials and structural tests for model development, calibration and validation. Most discussion concerns polymeric crushable foam, metal foam and metallic lattice structures.     

Some theoretical considerations on the dynamic response of sandwich structures under impulsive loading

A theoretical solution is obtained to predict the dynamic response of peripherally clamped square metallic sandwich panels with either honeycomb core or aluminium foam core under blast loading. In the theoretical analysis, the deformation of sandwich structures is separated into three phases, corresponding to the transfer of impulse to the front face velocity, core crushing and overall structural bending/stretching, respectively. The cellular core is assumed to have a progressive crushing deformation mode in the out-of-plane direction, with a dynamically enhanced plateau stress (for honeycombs). The in-plane strength of the cellular core is assumed unaffected by the out-of-plane compression. By adopting an energy dissipation rate balance approach developed by earlier researchers for monolithic square plates, but incorporating a newly developed yield condition for the sandwich panels in terms of bending moment and membrane force, “upper” and “lower” bounds are obtained for the maximum permanent deflections and response time. Finally, comparative studies are carried out to investigate: (1) influence of the change in the in-plane strength of the core after the out-of-plane compression; (2) performances of a square monolith panel and a square sandwich panel with the same mass per unit area; and (3) analytical models of sandwich beams and circular and square sandwich plates.     

冲击载荷作用下夹芯板的后屈曲研究

利用ABAQUS有限元软件,对SHPB作用下的夹芯板的变形特性进行了分析,将有限元计算结果和实验测量结果作对比。在冲击载荷时间历程内,着重从芯板的位移和应力以及应变这3个方面对芯板的后屈曲问题进行了分析,研究了夹芯板中弹塑性压缩波的传播过程,芯板在压缩波作用下的动力屈曲产生、传播特性,压缩波与屈曲变形的相互作用以及塑性后屈曲大变形的发展规律。     

Multiscale aggregating discontinuities: A method for circumventing loss of material stability

New methods for the analysis of failure by multiscale methods that invoke unit cells to obtain the subscale response are described. These methods, called multiscale aggregating discontinuities, are based on the concept of ‘perforated’ unit cells, which exclude subdomains that are unstable, i.e. exhibit loss of material stability. Using this concept, it is possible to compute an equivalent discontinuity at the coarser scale, including both the direction of the discontinuity and the magnitude of the jump. These variables are then passed to the coarse‐scale model along with the stress in the unit cell. The discontinuity is injected at the coarser scale by the extended finite element method. Analysis of the procedure shows that the method is consistent in power and yields a bulk stress–strain response that is stable. Applications of this procedure to crack growth in heterogeneous materials are given. Copyright © 2007 John Wiley & Sons, Ltd.     

新型负泊松比多孔吸能盒平台区力学性能

提出了一种具有负泊松比效应的汽车前纵梁吸能盒(NPRC)结构,通过对元胞平台区的失效模式和平台应力的分析,研究了此结构在失效时的力学性能,即等效弹性模量和平台应力在面内加载过程中均能得到一定程度的增强,表现出较好的能量吸收能力。根据NPRC元胞在平台区的力学模型,分别建立了发生弹性屈曲和塑性塌陷时的临界应力公式,得出塑性塌陷是该结构的主要失效模式。通过Matlab程序建立了NPRC元胞的参数化有限元模型,研究了元胞几何参数与平台应力的关系,即元胞的平台应力与长度系数和元胞夹角呈反比,与厚度系数呈正比。通过NPRC结构3×3样件的面内轴向准静态压缩实验验证了有限元分析结果,实验结果表明:NPRC样件等效负泊松比为-11.97,产生密实化现象,平台应力的峰值随着应变的增加逐渐增大,这对提高能量吸收性能具有重要的研究意义。     

Multifield modelling and failure prediction of cellular cores produced by selective laser melting

A multifield simulation approach of cellular cores produced by additive manufacturing is presented. The analysis is aiming to derive the relation between the manufacturing process parameters and the resulting material failure behaviour. To this purpose, the selective laser melting manufacturing process is initially thermo‐mechanically simulated, followed by the mechanical analysis of the nonlinear core behaviour. The methodology is demonstrated in the case of open‐lattice body‐centred‐cubic (BCC) cellular cores.     

Effects of particle plasticity characteristics on local interface stress in particle reinforced composite during uniaxial tension

For elastoplastic particle reinforced metal matrix composites, failure may originate from interface debonding between the particles and the matrix, both elastoplastic and matrix fracture near the interface. To calculate the stress and strain distribution in these regions, a single reinforcing particle axisymmetric unit cell model is used in this article. The nodes at the interface of the particle and the matrix are tied. The development of interfacial decohesion is not modelled. Finite element modelling is used, to reveal the effects of particle strain hardening rate, yield stress and elastic modulus on the interfacial traction vector (or stress vector), interface deformation and the stress distribution within the unit cell, when the composite is under uniaxial tension. The results show that the stress distribution and the interface deformation are sensitive to the strain hardening rate and the yield stress of the particle. With increasing particle strain hardening rate and yield stress, the interfacial traction vector and internal stress distribution vary in larger ranges, the maximum interfacial traction vector and the maximum internal stress both increase, while the interface deformation decreases. In contrast, the particle elastic modulus has little effect on the interfacial traction vector, internal stress and interface deformation.     

Elastic buckling of hexagonal chiral cell honeycombs

The paper describes a combined analytical, numerical and experimental analysis on the compressive strength of hexagonal chiral honeycombs due to elastic buckling of the unit cells under flatwise compressive loading. Hexagonal chiral honeycombs are cellular structures composed noncentresymmetric unit cells, with an in-plane negative Poisson’s ratio (NPR) with a value of −1. Cylinders connected by tangent ligaments at 60° degrees compose the unit cells. Approximated analytical models are proposed for the purpose of initial design assuming the main contribution to the elastic collapse stress being given by the nodes, and considering also the superposition of the critical elastic loads of each component of the unit cell. The models are expressed in terms of nondimensional geometric unit cell parameters (ligament to cylinder radius aspect ratio and relative density), and core material properties. Finite element calculations using shell and brick elements are also performed on unit cell models with periodic boundary conditions using linear bifurcation buckling analysis. The analytical and numerical results are compared with the outcome of a series of experimental flatwise compressive tests carried out on chiral honeycomb samples manufactured using rapid prototyping technique in PA sintered powder and ABS plastics. The comparison shows good convergence between the sets of results, and highlights the specific deformation mechanisms of the hexagonal chiral honeycomb cell.     

Scale‐related topology optimization of cellular materials and structures

The integrated optimization of lightweight cellular materials and structures are discussed in this paper. By analysing the basic features of such a two‐scale problem, it is shown that the optimal solution strongly depends upon the scale effect modelling of the periodic microstructure of material unit cell (MUC), i.e. the so‐called representative volume element (RVE). However, with the asymptotic homogenization method used widely in actual topology optimization procedure, effective material properties predicted can give rise to limit values depending upon only volume fractions of solid phases, properties and spatial distribution of constituents in the microstructure regardless of scale effect. From this consideration, we propose the design element (DE) concept being able to deal with conventional designs of materials and structures in a unified way. By changing the scale and aspect ratio of the DE, scale‐related effects of materials and structures are well revealed and distinguished in the final results of optimal design patterns. To illustrate the proposed approach, numerical design problems of 2D layered structures with cellular core are investigated. Copyright © 2006 John Wiley & Sons, Ltd.     

Three-dimensional elastoplastic finite element modelling of deformation and fracture behaviour of rubber-modified polycarbonates at different triaxiality

A periodic face-centred cuboidal cell model is provided to account for inter-particle interaction, and a particle-crack tip interaction model is developed to study the interaction between a blunting model I crack tip and the closest array of initially spherical rubber particles in an effective medium. Three-dimensional elastoplastic finite element analysis has been preformed to study the deformation and fracture behaviour of rubber-modified polycarbonates. The effective elastoplastic constitutive relation is derived by the method of homogenisation and local stress and strain distributions are obtained to explore the role of rubber cavitation in the toughening process at different stress triaxiality. 3D elastoplastic finite element results are compatible with experimental observations, that is, rubber particles can act as stress concentrators to initiate crazing or shear yielding in the matrix but they behave differently from voids at high triaxiality. Rubber cavitation plays an important role in the toughening process under high tensile triaxial stresses.     

A unified mathematical programming formulation of strain driven and interior point algorithms for shakedown and limit analysis

A mathematical programming formulation of strain‐driven path‐following strategies to perform shakedown and limit analysis for perfectly elastoplastic materials in an FEM context is presented. From the optimization point of view, standard arc‐length strain‐driven elastoplastic analyses, recently extended to shakedown, are identified as particular decomposition strategies used to solve a proximal point algorithm applied to the static shakedown theorem that is then solved by means of a convergent sequence of safe states. The mathematical programming approach allows: a direct comparison with other non‐linear programming methods, simpler convergence proofs and duality to be exploited. Owing to the unified approach in terms of total stresses, the strain‐driven algorithms become more effective and less non‐linear with respect to a self‐equilibrated stress formulation and easier to implement in the existing codes performing elastoplastic analysis. The elastic domain is represented avoiding any linearization of the yield function so improving both the accuracy and the performance. Better results are obtained using two different finite elements, one with a good behavior in the elastic range and the other suitable for performing elastoplastic analysis. The proposed formulation is compared with a specialized implementation of the primal–dual interior point method suitable to solve the problems at hand. Copyright © 2011 John Wiley & Sons, Ltd.     

Stress–strain curves of metallic materials and post‐necking strain hardening characterization: A review

For metallic materials, standard uniaxial tensile tests with round bar specimens or flat specimens only provide accurate equivalent stress–strain curve before diffuse necking. However, for numerical modelling of problems where very large strains occur, such as plastic forming and ductile damage and fracture, understanding the post‐necking strain hardening behaviour is necessary. Also, welding is a highly complex metallurgical process, and therefore, weldments are susceptible to material discontinuities, flaws, and residual stresses. It becomes even more important to characterize the equivalent stress–strain curve in large strains of each material zone in weldments properly for structural integrity assessment. The aim of this paper is to provide a state‐of‐the‐art review on quasi‐static standard tensile test for stress–strain curves measurement of metallic materials. Meanwhile, methods available in literature for characterization of the equivalent stress–strain curve in the post‐necking regime are introduced. Novel methods with axisymmetric notched round bar specimens for accurately capturing the equivalent stress–strain curve of each material zone in weldment are presented as well. Advantages and limitations of these methods are briefly discussed.     

Mechanical behavior of sandwich panels with hollow Al–Si tubes core construction

A new type of lightweight sandwich panels consisting of vertically aligned hollow Al–Si alloy tubes as core construction and carbon fiber composite face sheets was designed. The hollow Al–Si alloy tubes were fabricated using precision casting and were bonded to the face sheets using an epoxy adhesive. The out-of-plane compression (i.e. core crushing), in-plane compression, and three-point bending response of the panels were tested until failure. The hollow Ai–Si alloy tubes core configuration show superior specific strength under crushing compared to common metallic and stochastic foam cores. Under in-plane compression and three-point bending, the buckling of face sheets and debonding of hollow cores from the face sheets were observed. Simple analytical relationships based on the concepts of mechanics of materials were provided for the compression tests, which estimate the sandwich panels’ strength with high fidelity. For three-point bending, detailed finite element analysis was used to model the response and initial failure of the sandwich panels.     

Effect of low temperature on plastic deformation in structural materials in a complex state of stress

The results of experimental studies of elastoplastic deformation in structural materials of different grades in a complex state of stress at low temperatures are analyzed. It is shown that the degree of the effect of temperature and force conditions on the character of materials’ strain hardening at low temperature depends on their nature and structural state. The principal hypotheses and postulates that form the basis of the modern plasticity theories are examined for low temperatures. Two new models are introduced to describe the elastoplastic deformation of quasi-brittle and metastable materials under the simple loading with consideration for the effect of low temperatures on their structural state and mechanical properties.     

Mechanical characterization of hollow ceramic nanolattices

In the analysis of complex, hierarchical structural meta-materials, it is critical to understand the mechanical behavior at each level of hierarchy in order to understand the bulk material response. We report the fabrication and mechanical deformation of hierarchical hollow tube lattice structures with features ranging from 10 nm to 100 μm, hereby referred to as nanolattices. Titanium nitride (TiN) nanolattices were fabricated using a combination of two-photon lithography, direct laser writing, and atomic layer deposition. The structure was composed of a series of tessellated regular octahedra attached at their vertices. In situ uniaxial compression experiments performed in combination with finite element analysis on individual unit cells revealed that the TiN was able to withstand tensile stresses of 1.75 GPa under monotonic loading and of up to 1.7 GPa under cyclic loading without failure. During the compression of the unit cell, the beams bifurcated via lateral-torsional buckling, which gave rise to a hyperelastic behavior in the load–displacement data. During the compression of the full nanolattice, the structure collapsed catastrophically at a high strength and modulus that agreed well with classical cellular solid scaling laws given the low relative density of 1.36 %. We discuss the compressive behavior and mechanical analysis of the unit cell of these hollow TiN nanolattices in the context of finite element analysis in combination with classical buckling laws, and the behavior of the full structure in the context of classical scaling laws of cellular solids coupled with enhanced nanoscale material properties.     

Structurally Controlled Cellular Architectures for High‐Performance Ultra‐Lightweight Materials

The design and synthesis of cellular structured materials are of both scientific and technological importance since they can impart remarkably improved material properties such as low density, high mechanical strength, and adjustable surface functionality compared to their bulk counterparts. Although reducing the density of porous structures would generally result in reductions in mechanical properties, this challenge can be addressed by introducing a structural hierarchy and using mechanically reinforced constituent materials. Thus, precise control over several design factors in structuring, including the type of constituent, symmetry of architectures, and dimension of the unit cells, is extremely important for maximizing the targeted performance. The feasibility of lightweight materials for advanced applications is broadly explored due to recent advances in synthetic approaches for different types of cellular architectures. Here, an overview of the development of lightweight cellular materials according to the structural interconnectivity and randomness of the internal pores is provided. Starting from a fundamental study on how material density is associated with mechanical performance, the resulting structural and mechanical properties of cellular materials are investigated for potential applications such as energy/mass absorption and electrical and thermal management. Finally, current challenges and perspectives on high‐performance ultra‐lightweight materials potentially implementable by well‐controlled cellular architectures are discussed.