Design optimization of truss-cored sandwiches with homogenization

Lightweight metallic sandwich plates comprising periodic truss cores and solid facesheets are optimally designed against minimum weights. Constitutive models of the truss core are developed using homogenization techniques which, together with effective single-layer sandwich approaches, form the basis of a two-dimensional (2D) single-layer sandwich model. The 2D model is employed to simulate the mechanical behaviors of truss-cored sandwich panels having a variety of core topologies. The types of loading considered include bending, transverse shear and in-plane compression. The validities of the 2D model predictions are checked against direct FE simulations on three-dimensional (3D) truss core sandwich structures. Optimizations using the 2D sandwich model are subsequently performed to determine the minimum weights of truss-cored sandwiches subjected to various failure constraints: overall and local buckling, yielding and facesheet wrinkling. The performances of the optimized truss core sandwiches with 4-rod unit cell and solid truss members and pyramidal unit cell with hollow truss members are compared with benchmark lightweight structures such as honeycomb-cored sandwiches, tetrahedral core sandwiches and hat-stiffened single layer plates.     

Three-dimensional Composite Lattice Structures Fabricated by Electrical Discharge Machining

We present a novel method for fabricating carbon fiber composite sandwich panels with lattice core construction by means of electrical discharge machining (EDM). First, flat-top corrugated carbon fiber composite cores were fabricated by a hot press molding method. Then, two composite face sheets were bonded to each corrugated core to create precursor sandwich panels. These panels were transformed into sandwich panels with near-pyramidal truss cores by EDM plunge-cutting the corrugated core between the face sheets with a shaped cuprite electrode. The flat top corrugation permits adhesive to be applied consistently, and the selected dimensions leave a substantial bond area after cutting, resulting in a strong core-to-sheet bond. The crushing behavior of this novel construction was investigated in flatwise compression, and the results were compared to analytical expressions for strength and stiffness.     

金字塔栅格夹心夹层板动力响应分析

本文将金字塔形栅格夹心夹层板假设成均匀夹心夹层板，应用Reissner夹层板理论，对其自振频率以及在简谐荷载下的强迫振动进行了研究，并以简支板为例，得到其解析解，通过与有限元分析进行比较，两者结果吻合良好。并把金字塔形栅格夹层板与同质量实体板进行比较，得出金字塔形栅格夹层板具有更好动力性能。     

Dynamic compression of metallic sandwich structures during planar impulsive loading in water

The compressive response of rigidly supported stainless steel sandwich panels subject to a planar impulsive load in water is investigated. Five core topologies that spanned a wide range of crush strengths and strain-dependencies were investigated. They included a (i) square-honeycomb, (ii) triangular honeycomb, (iii) multi-layer pyramidal truss, (iv) triangular corrugation and (v) diamond corrugation, all with a core relative density of approximately 5%. Quasi-statically, the honeycombs had the highest peak strength, but exhibited strong softening beyond the peak strength. The truss and corrugated cores had significantly lower strength, but a post yield plateau that extended to beyond a plastic strain of 60% similar to metal foams. Dynamically, the transmitted pressures scale with the quasi-static strength. The final transmitted momentum increased slowly with core strength (provided the cores were not fully crushed). It is shown that the essential aspects of the dynamic response, such as the transmitted momentum and the degree of core compression, are captured with surprising fidelity by modeling the cores as equivalent metal foams having plateau strengths represented by the quasi-static peak strength. The implication is that, despite considerable differences in core topology and dynamic deformation modes, a simple foam-like model replicates the dynamic response of rigidly supported sandwich panels subject to planar impulsive loads. It remains to ascertain whether such foam-like models capture more nuanced aspects of sandwich panel behavior when locally loaded in edge clamped configurations.     

Theoretical prediction on corrugated sandwich panels under bending loads

In this paper, an aluminum corrugated sandwich panel with triangular core under bending loads was investigated. Firstly, the equivalent material parameters of the triangular corrugated core layer, which could be considered as an orthotropic panel, were obtained by using Castigliano’s theorem and equivalent homogeneous model. Secondly, contributions of the corrugated core layer and two face panels were both considered to compute the equivalent material parameters of the whole structure through the classical lamination theory, and these equivalent material parameters were compared with finite element analysis solutions. Then, based on the Mindlin orthotropic plate theory, this study obtain the closed-form solutions of the displacement for a corrugated sandwich panel under bending loads in specified boundary conditions, and parameters study and comparison by the finite element method were executed simultaneously.     

Minimum weights of pressurized hollow sandwich cylinders with ultralight cellular cores

Long, open-ended, hollow sandwich cylinders with ultralightweight cellular cores are optimized under uniform internal pressure for minimum weight design. Five different core topologies are considered: Kagomé truss, single-layered pyramidal truss, double-layered pyramidal truss, single-layered corrugated core and double-layered corrugated core. The highly porous cellular materials are homogenized to obtain effective constitutive relations. Close-formed solutions are presented for the forces and stresses in individual structural members of the sandwich, which are then validated by finite element calculations. Optimization of the sandwich-walled hollow cylinder is achieved using a quadratic optimizer, subjected to the constraints that none of the following failure modes occurs: facesheet yielding; facesheet punch shearing (active only for truss-cored sandwiches); core member buckling; core member yielding. In comparison with hollow cylinders having solid walls, truss-core sandwich cylinders and single-layer corrugated core sandwich cylinders are found to have superior weight advantages, especially for more heavily loaded cases. With the consideration of both weight efficiency and failure modes, sandwich-walled hollow cylinders having Kagomé truss core with pyramidal sub-geometry have the best overall performance in comparison with other core topologies.     

新型复合材料点阵结构的研究进展

复合材料点阵结构是一种具有轻质、高比强、高比刚以及多功能潜力的新型结构材料, 近几年受到国外学者的极大关注, 是新一代结构材料一体化的理想结构材料. 本文概述了点阵复合材料及结构的发展历程, 包括复合材料点阵结构的拓扑构型设计、制备工艺研究、力学性能表征、失效模式分析、预报模型评价等方面的工作, 并给出了复合材料点阵结构的力学性能、失效模式和理论数值模型汇总表以及修正后的材料强度与密度关系图. 同时, 本文对复合材料点阵结构可能应用的领域进行预测, 并对其未来发展进行了展望.      

The impulsive response of sandwich beams: Analytical and numerical investigation of regimes of behaviour

An analytical model is developed to classify the impulsive response of sandwich beams based on the relative time-scales of core compression and the bending/stretching response of the sandwich beam. It is shown that an overlap in time scales leads to a coupled response and to the possibility of an enhanced shock resistance. Four regimes of behaviour are defined: decoupled responses with the sandwich core densifying partially or completely, and coupled responses with partial or full core densification. These regimes are marked on maps with axes chosen from the sandwich beam transverse core strength, the sandwich beam aspect ratio and the level of blast impulse. In addition to predicting the time-scales involved in the response of the sandwich beam, the analytical model is used to estimate the back face deflection, the degree of core compression and the magnitude of the support reactions. The predictions of the analytical model are compared with finite element (FE) simulations of impulsively loaded sandwich beams comprising an anisotropic foam core and elastic, ideally plastic face-sheets. The analytical and numerical predictions are in good agreement up to the end of core compression. However, the analytical model under-predicts the peak back face deflection and over-predicts the support reactions, especially for sandwich beams with high strength cores. The FE calculations are employed to construct design charts to select the optimum transverse core strength that either minimises the back face deflections or support reactions for a given sandwich beam aspect ratio or blast impulse. Typically, the value of the transverse core strength that minimises the back face deflection also minimises the support reactions. However, the optimal core strength depends on the level of blast impulse, with higher strength cores required for greater blasts.     

Elastic structural response of prismatic metal sandwich tubes to internal moving pressure loading

The analytical and numerical modeling of the structural response of a prismatic metal sandwich tube subjected to internal moving pressure loading is investigated in this paper. The prismatic core is equivalent to homogeneous and cylindrical orthotropic solids via homogenization procedure. The sandwich tube with the “effective” homogenized core is modeled using multi-layer sandwich theory considering the effects of transverse shear deformation and compressibility of the core; moreover, the solutions are obtained by using the precise integration method. Several dynamic elastic finite element (FE) simulations are carried out to obtain the structural response of the tube to shock loading moving at different velocities. The comparison between analytic solutions and FE simulations demonstrates that the transient analytical model, based on the proposed sandwich model, is capable of predicting the critical velocity and the dynamic structural response of the sandwich tube with the “effective” homogenized core with a high degree of accuracy. In addition, the critical velocity predicted using FE simulations of the complete model is not in agreement with that of the effective model. However, the structural response and the maximum amplification factors obtained using FE simulations of the complete model are nearly similar to that of the effective model, when the shock loading moves at the critical velocity. The influences of the relative density on the structural response are studied, and the capabilities of load bearing for sandwich tubes with different cores are compared with each other and with the monolithic tube. The results indicate that Kagome and triangle-6 are preferred among five topologies.     

层级褶皱结构夹层板的等效弹性常数

针对二级层级褶皱结构夹层板,通过变形协调研究了其等效弹性常数。首先对一级层级褶皱结构进行正交各向异性等效,得到一级等效弹性常数;将二级层级褶皱结构看成是由正交各向异性材料组成的三角形桁架夹心,将二级层级褶皱结构等效为均匀连续正交各向异性板,依据夹层板面板与夹心变形协调特点得到夹层板整体等效弹性常数。结合结构几何参数对等效公式的误差进行了讨论,并对等效公式做出修正。通过与数值分析结果对比,表明本文提出的等效公式具有较高精度。     

碳纤维复合材料金字塔点阵结构制备工艺及力学性能研究

本文针对碳纤维复合材料点阵结构,从结构设计、制备工艺、平压性能、剪切性能等方面对其进行试验表征及理论模型研究.设计四种成型碳纤维复合材料金字塔点阵结构的思想,并采用一种新的制备工艺即预浸料二次成型工艺制备试样,试验结果表明,该工艺能最大程度发挥纤维增强潜力.通过实验揭示在平压载荷下杆件屈曲、杆件断裂、杆件分层脱胶失效机理,在剪切载荷下杆件屈曲、杆件分层、杆件脱胶失效机理,基于结构力学基础原理,建立相应理论模型,经过修正之后的理论模型均能较好预报典型载荷下力学性能.本文研究发现碳纤维复合材料金字塔点阵结构具有密度低、比强度大、比刚度高等优点,且芯子中具有大量空间,可以制备轻质多功能结构.     

Imperfection sensitivity of pyramidal core sandwich structures

Lightweight metallic truss structures are currently being investigated for use within sandwich panel construction. These new material systems have demonstrated superior mechanical performance and are able to perform additional functions, such as thermal management and energy amelioration. The subject of this paper is an examination of the mechanical response of these structures. In particular, the retention of their stiffness and load capacity in the presence of imperfections is a central consideration, especially if they are to be used for a wide range of structural applications. To address this issue, sandwich panels with pyramidal truss cores have been tested in compression and shear, following the introduction of imperfections. These imperfections take the form of unbound nodes between the core and face sheets—a potential flaw that can occur during the fabrication process of these sandwich panels. Initial testing of small scale samples in compression provided insight into the influence of the number of unbound nodes but more importantly highlighted the impact of the spatial configuration of these imperfect nodes. Large scale samples, where bulk properties are observed and edge effects minimized, have been tested. The stiffness response has been compared with finite element simulations for a variety of unbound node configurations. Results for fully bound cores have also been compared to existing analytical predictions. Experimentally determined collapse strengths are also reported. Due to the influence of the spatial configuration of unbound nodes, upper and lower limits on stiffness and strength have been determined for compression and shear. Results show that pyramidal core sandwich structures are robust under compressive loading. However, the introduction of these imperfections causes rapid degradation of core shear properties.     

四边简支正交各向异性蜂窝型夹层板固有频率计算

该文以四边简支的方形蜂窝矩形夹层板为例,在经典夹层板理论的基础上,运用离散结构形式的运动控制方程和线性微分算子的可交换性,给出了一种把具有蜂窝型夹心的夹层板的包含三个广义位移的控制方程组化为,仅包含一个广义位移函数的单一方程的简单方法,并给出了四边简支蜂窝型夹层板的固有频率的精确解。研究结果对蜂窝夹层板的结构设计和工程应用具有指导意义。     

Design of metallic textile core sandwich panels

Metallic sandwich panels with textile cores have been analyzed subject to combined bending and shear and then designed for minimum weight. Basic results for the weight benefits relative to solid plates are presented, with emphasis on restricted optimizations that assure robustness (non-catastrophic failure) and acceptable thinness. Select numerical simulations are used to check the analytical results and to explore the role of strain hardening beyond failure initiation. Comparisons are made with competing concepts, especially honeycomb and truss core systems. It is demonstrated that all three systems have essentially equivalent performance. The influence on the design of a concentrated compressive stress that might crush the core has been explored and found to produce relatively small effect over the stress range of practical interest. “Angle ply” cores with members in the ±45° orientation are found to be near optimal for all combinations of bending, shear and compression.     

Modeling of the role of defects in sintered FeCrAlY foams

The metal sintering approach offers a cost- effective means for the mass-production of open-cell foams from a range of materials, including high-temperature steel alloys, which offer novel mechanical and acoustic properties. In a separate experimental study, the mechanical properties of open-celled steel alloy (FeCrAlY) foams have been characterized under uniaxial compression and shear loading. Compared to predictions from established models, a significant knockdown in material properties was observed. This knockdown was attributed to the presence of defects throughout the microstructure that result from the unique fabrication process. In the present paper, the microstructure of sintered FeCrAlY foams was modeled by using a finite element (FE) model. In particular, microstructural variations were introduced to a base lattice, and the effects on the strength and stiffness calculated. A range of defects identified under scanning electronic microscope (SEM) imaging were considered including broken ligaments, thickness variations, and pore blockages, which are the three primary imperfections observed in sintered foams. The corresponding levels of defect present in the material were subsequently input into the FE model, with the resulting predictions correlating well with experimental data. The project supported by the National Basic Research Program of China (2006CB601202), the National Natural Science Foundation of China (10328203, 10572111, 10632060), the National 111 Project of China (B06024), and the US Office of Naval Research (N000140210117).     

轻质金属点阵夹层板热屈曲临界温度分析

本文针对均匀温度场下四边简支和四边固支金属点阵夹层板的临界热屈曲温度进行了求解和参数影响分析。将点阵夹芯等效为均匀连续体,并且将夹层板的剪切刚度近似为点阵夹芯的抗剪切刚度,忽略夹芯的抗弯刚度且认为夹层板主要由面板来提供抗弯刚度。对于无法获得解析解的四边固支条件,通过对未知变量进行双傅里叶展开的方法求解了Ressiner夹层板模型的临界屈曲温度,理论分析结果与有限元计算结果吻合良好。进一步分析了不同边界条件、点阵胞元构型、点阵材料相对密度、面板厚度等对临界屈曲温度的影响规律。     

Strength optimization of metallic sandwich panels subject to bending

A general methodology for the design of strong, lightweight sandwich panels is described and implemented. Several core topologies are considered, including square-section truss members in pyramidal and tetrahedral configurations, square honeycombs, and corrugated sheets. When the number of independent design parameters is restricted to three, closed-form analytical solutions for the optimal design are obtained. Alternatively, when a fourth parameter is added (as needed to fully characterize the panel geometry), numerical routes are required. The results demonstrate that the three parameter optimizations yield design weights that are only slightly heavier than those of the fully optimized panels, provided the value of the fourth parameter is selected judiciously. The weight rankings of the various core topologies change with load capacity, although the differences between them are generally small, particularly upon comparison with the weight of a solid panel.     

Using strain energy-based prediction of effective elastic properties in topology optimization of material microstructures

An alternative strain energy method is proposed for the prediction of effective elastic properties of orthotropic materials in this paper. The method is implemented in the topology optimization procedure to design cellular solids. A comparative study is made between the strain energy method and the well-known homogenization method. Numerical results show that both methods agree well in the numerical prediction and sensitivity analysis of effective elastic tensor when homogeneous boundary conditions are properly specified. Two dimensional and three dimensional microstructures are optimized for maximum stiffness designs by combining the proposed method with the dual optimization algorithm of convex programming. Satisfactory results are obtained for a variety of design cases. The project supported by the National Natural Science Foundation of China (10372083, 90405016), 973 Program (2006CB601205) and the Aeronautical Science Foundation (04B53080). The English text was polished by Keren Wang.     

Boundary element analysis for elastic and elastoplastic problems of 2D orthotropic media with stress concentration

Both the orthotropy and the stress concentration are common issues in modern structural engineering. This paper introduces the boundary element method (BEM) into the elastic and elastoplastic analyses for 2D orthotropic media with stress concentration. The discretized boundary element formulations are established, and the stress formulae as well as the fundamental solutions are derived in matrix notations. The numerical procedures are proposed to analyze both elastic and elastoplastic problems of 2D orthotropic media with stress concentration. To obtain more precise stress values with fewer elements, the quadratic isoparametric element formulation is adopted in the boundary discretization and numerical procedures. Numerical examples show that there are significant stress concentrations and different elastoplastic behaviors in some orthotropic media, and some of the computational results are compared with other solutions. Good agreements are also observed, which demonstrates the efficiency and reliability of the present BEM in the stress concentration analysis for orthotropic media. The project supported by the Basic Research Foundation of Tsinghua University, the National Foundation for Excellent Doctoral Thesis (200025) and the National Natural Science Foundation of China (19902007). The English text was polished by Keren Wang.     

Wrinkling of thick orthotropic sandwich plates under general loading conditions

Summary  This contribution presents an efficient analytical model as well as a FE computation of the critical load, which leads to local stability failure (wrinkling) in sandwich structures. The analytical model assumes an orthotropic face layer and a thick transversely isotropic core. In the last section, a more general core material model is considered. Common core materials (foams and honeycombs) can be described with good accuracy within this model. The main advantage of the solution is the consideration of general loading conditions for the orthotropic face layer as well as in-plane deformations of the core. The results of the FE calculations and the analytical model are in good agreement with each other. Received 7 January 1999; accepted for publication 15 June 1999