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.     

Underwater blast loading of sandwich beams: Regimes of behaviour

Finite element (FE) calculations are used to develop a comprehensive understanding of the dynamic response of sandwich beams subjected to underwater blast loading, including the effects of fluid–structure interaction. Design maps are constructed to show the regimes of behaviour over a broad range of loading intensity, sandwich panel geometry and material strength. Over the entire range of parameters investigated, the time-scale associated with the initial fluid–structure interaction phase up to the instant of first cavitation in the fluid is much smaller than the time-scales associated with the core compression and the bending/stretching responses of the sandwich beam. Consequently, this initial fluid–structure interaction phase decouples from the subsequent phases of response. Four regimes of behaviour exist: the period of sandwich core compression either couples or decouples with the period of the beam bending, and the core either densifies partially or fully. These regimes of behaviour are charted on maps using axes of blast impulse and core strength. The simulations indicate that continued loading by the fluid during the core compression phase and the beam bending/stretching phase cannot be neglected. Consequently, analyses that neglect full fluid–structure interaction during the structural responses provide only estimates of performance metrics such as back face deflection and reaction forces at the supports. The calculations here also indicate that appropriately designed sandwich beams undergo significantly smaller back face deflections and exert smaller support forces than monolithic beams of equal mass. The optimum transverse core strength is determined for minimizing the back face deflection or support reactions at a given blast impulse. Typically, the transverse core strength that minimizes back face deflection is 40% below the value that minimizes the support reaction. Moreover, the optimal core strength depends upon the level of blast impulse, with higher strength cores required for higher intensity blasts.     

The response of clamped sandwich plates with lattice cores subjected to shock loading

The dynamic response of clamped circular monolithic and sandwich plates of equal areal mass and thickness has been measured by loading the plates at mid-span with metal foam projectiles. The sandwich plates comprise AISI 304 stainless steel face sheets and either AL-6XN stainless steel pyramidal core or AISI 304 stainless steel square-honeycomb lattice core. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the plates as a function of projectile momentum. It is found that the sandwich plates have a higher shock resistance than monolithic plates of equal mass, and the square-honeycomb sandwich plates outperform the pyramidal core plates. Three-dimensional finite element simulations of the experiments are in good agreement with the experimental measurements. The finite element calculations indicate that the ratio of loading time to structural response time is approximately 0.5. Consequently, the tests do not lie in the impulsive regime, and projectile momentum alone is insufficient to quantify the level of loading.     

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.     

Finite element analysis of the dynamic response of clamped sandwich beams subject to shock loading

The finite element (FE) method is employed to analyse the response of clamped sandwich beams subject to shock loadings. Pressure versus time histories representative of shock loadings are applied uniformly to the outer face of the sandwich beam; an impulse applied uniformly to the outer face of the sandwich beam is shown to model adequately shock loadings. Material elasticity and strain hardening representative of structural steels have only a minor effect upon the beam response. Further, the magnitude of the compressive strength of the core has only a limited influence upon the dynamic response of the sandwich beam for the representative range of core strengths considered. The FE results for the deflections and structural response time agree well with the rigid ideally-plastic analytical predictions of Fleck and Deshpande (J. Appl. Mech. (2003), in press).     

Optimal design of box-section sandwich beams in three-point bending

Minimum mass designs are obtained for box-section sandwich beams of various cross-sections in three-point bending. The overall compliance of the hollow, tubular beams are decomposed additively into a global contribution due to macroscopic bending (Timoshenko beam theory) and a local contribution associated with transverse deflection of the walls of the hollow beam adjacent to the central loading patch. The structural response is analysed for beams of square sections with various internal topologies: a solid section, a foam-filled tube with monolithic walls, a hollow tube with walls made from sandwich plates, and a hollow tube with walls reinforced by internal stiffeners. Finite element analysis is used to validate analytical models for the overall stiffness of the tubes in three-point bending. Minimum mass designs are obtained as a function of the overall stiffness, and the relative merits of the competing topologies are discussed.     

One-dimensional response of sandwich plates to underwater shock loading

The one-dimensional shock response of sandwich plates is investigated for the case of identical face sheets separated by a compressible foam core. The dynamic response of the sandwich plates is analysed for front face impulsive loading, and the effect of strain hardening of the core material is determined. For realistic ratios of core mass to face sheet mass, it is found that the strain hardening capacity of the core has a negligible effect upon the average through-thickness compressive strain developed within the core. Consequently, it suffices to model the core as an ideally plastic-locking solid. The one-dimensional response of sandwich plates subjected to an underwater pressure pulse is investigated by both a lumped parameter model and a finite element (FE) model. Unlike the monolithic plate case, cavitation does not occur at the fluid-structure interface, and the sandwich plates remain loaded by fluid until the end of the core compression phase. The momentum transmitted to the sandwich plate increases with increasing core strength, suggesting that weak sandwich cores may enhance the underwater shock resistance of sandwich plates.     

爆炸载荷作用下具有可折叠芯层夹芯梁的动态响应

基于目前研究最广泛的刚性折纸(Tachi-origami)样式,通过改变其初始折叠角度构建出4种不同的蜂窝胞元,并且通过排列分布将其组成夹芯梁。采用商用有限元软件Abaqus/explicit对准静态和爆炸载荷作用下可折叠芯层夹芯梁的力学响应进行研究,分析可折叠芯层的泊松比变化规律、夹芯梁背板挠度以及能量吸收机理;并将夹芯梁与等质量的实体梁进行对比。采用后面板最大挠度作为抗爆性能的评价,结果发现:可折叠芯层在准静态载荷下具有一定的负泊松比效应;夹芯梁的抗爆性能优于实体梁,曲边蜂窝的初始折角对其作为芯层夹芯梁的抗爆性能有较大影响,随着初始折角的逐渐增大,其抗爆性能逐渐下降;当初始折角为直角时对应于方孔直边蜂窝,其抗爆性能最差。     

Test method for measuring strength of a curved sandwich beam

A fixture for testing curved sandwich beams in flexure was designed and evaluated. The test specimen is a continuous sandwich beam consisting of a central circular 90° region connected by two straight legs. The fixture was designed according to the four-point flexure principle to produce a pure bending moment in the curved region. The validity of the test fixture in producing the desired loading was examined by fitting a curved aluminum bar of similar bending stiffness as the sandwich beams considered. Strain gage readings were successfully compared to predictions from curved homogeneous beam theory. In addition, the deflection of the beam at the loading points was analyzed using straight and curved beam theory for the various sections of the beam, and predictions were compared to measured load-displacement response. Good agreement was achieved between experimental and analytical results lending confidence to the test principle. Curved sandwich beams consisting of glass/polyester face sheets over a PVC foam core were tested to failure and the loading response of the beams and their failure behavior are discussed. It was found that the beams failed at the upper face/core interface due to radial tension stress.     

高速冲击载荷下梯度金属泡沫夹芯梁的动态响应与失效

采用泡沫弹冲击加载实验对梯度金属泡沫夹芯梁结构开展了不同冲击强度下的动态响应和失效研究,分析了由三种不同密度泡沫铝组成的等面密度的五种不同梯度的夹芯结构在夹支边界条件下的抗高速冲击性能,结合三点弯曲实验,研究梯度效应对夹芯结构抗冲击性能的影响。研究表明：密度梯度对结构的失效过程和失效模式有着明显的影响,且夹芯梁结构的初始失效模式对结构整体响应和主要的能量吸收机制起着主导作用;当冲击条件不足以使得均质芯材发生压缩时,均质及负梯度夹芯结构初始失效模式为整体弯曲变形,低强度芯层位于前两层的梯度结构随着冲击强度的变化出现不同程度的局部芯层压缩;当冲击强度较低时,梯度结构通过丰富的局部失效表现出明显优于均质结构的抗冲击变形能力;当冲击强度大于临界值时,均质结构具有更好的抗冲击变形能力。通过合理地设计密度梯度实现逐层压缩吸能,能够有效的提升防护结构的抗冲击性能。     

The dynamic response of composite sandwich beams to transverse impact

The dynamic response of glass fibre–vinylester composite beams is measured by impacting the beams at mid-span with metal foam projectiles. The beams exist in composite monolithic form, and in sandwich configuration with composite face-sheets and a core made from PVC foam or end-grain balsa wood. High-speed photography is used to measure the transient transverse deflection of the beams and to record the dynamic modes of deformation and failure. For both monolithic and sandwich configurations, a flexural wave travels from the impact site towards the supports. Ultimate failure of the monolithic and sandwich beams is by tensile tearing of the faces. The sandwich beams also exhibit cracking of the core, and face-sheet delamination. The dynamic strength of the beams is quantified by the maximum transient transverse deflection at mid-span of the beams as a function of projectile momentum. It is demonstrated that sandwich beams can outperform monolithic beams of equal mass. The trade-off between core strength and core thickness is such that a low density PVC foam core outperforms a higher density PVC foam core. End-grain balsa wood has a superior stiffness and strength to that of PVC foam in compression and in shear. Consequently, sandwich beams with a balsa core outperform beams with a PVC foam core for projectiles of low momentum. The order reverses at high values of projectile momentum: the sandwich beams with a balsa wood core fail prematurely in longitudinal shear by splitting along the grain.     

Analytical modeling and finite element simulation of the plastic collapse of sandwich beams with pin-reinforced foam cores

Analytical predictions are presented for the plastic collapse strength of lightweight sandwich beams having pin-reinforced foam cores that are loaded in 3-point bending. Both polymer and aluminum foam cores are considered, whilst the facesheet and the pins are made of either composite or metal. Four different failure modes are account for: metal facesheet yield or composite facesheet microbuckling, facesheet wrinkling, plastic shear of the core, and facesheet indentation beneath the loading rollers. A micromechanics-based model is developed and combined with the homogenization approach to calculate the effective properties of pin-reinforced foam cores. To calculate the elastic buckling strength of pin reinforcements, the pin-reinforced foam core is treated as assemblies of simply supported columns resting upon an elastic foundation. Minimum mass design of the sandwich is then obtained as a function of the prescribed structural load index, subjected to the constraint that none of the above failure modes occurs. Collapse mechanism maps are constructed and compared with the failure maps of foam-cored sandwich beams without pin reinforcements. Finite element simulations are carried out to verify the analytical model and to study the performance and failure mechanisms of the sandwich subject to loading types other than 3-point bending. The results demonstrate that the weaker the foam is, the more optimal the pin-reinforced foam core becomes, and that sandwich beams with pin-reinforced polymer foam cores are structurally more efficient than foam- or truss-cored sandwich beams.     

撞击载荷下泡沫铝夹芯梁的塑性动力响应

应用泡沫金属子弹撞击加载的方式研究了固支泡沫铝夹芯梁和等质量实体梁的塑性动力响应。 采用激光测速装置和位移传感器测量了泡沫子弹的撞击速度和后面板中心点的位移-时间曲线,研究了加载 冲量、面板厚度和芯层厚度对夹芯梁抗冲击性能的影响。给出了泡沫铝夹芯梁的变形与失效模式,实验结果 表明结构响应对夹芯结构配置比较敏感,后面板中心点的残余变形与加载冲量、面板厚度呈线性关系。与等 质量实体梁的比较表明,泡沫铝夹芯梁具有更好的抗冲击能力。实验结果对多孔金属夹芯结构的优化设计具 有一定的参考价值。     

Performance of metallic honeycomb-core sandwich beams under shock loading

Stainless steel square honeycomb core sandwich and solid monolithic beams have been subjected to high-pressure, short-duration impulses using a shock simulation technique involving high-speed impact of Al foam projectiles. The experiments have been designed to achieve two objectives: (i) to demonstrate the benefits of sandwich construction, and (ii) to assess the fidelity of dynamic finite element calculations in simulating the structural response. The results affirm that, when subjected to impulse levels representative of those associated with nearby explosions, the sandwich beams exhibit smaller displacements than the solid beams at equivalent weight. The benefit is especially large at lower impulses where the effective dynamic strength of the honeycomb core prevents crushing. The measurements and finite element simulations having greatest relevance to the shock resistance are found to correspond closely, particularly the displacements and the core crushing strains. One implication is that the dynamic finite element model has the requisite fidelity at impulse levels of interest.     

Metallic sandwich panels subjected to multiple intense shocks

The mechanical response and fracture of metal sandwich panels subjected to multiple impulsive pressure loads (shocks) were investigated for panels with honeycomb and folded plate core constructions. The structural performance of panels with specific core configurations under multiple impulsive pressure loads is quantified by the maximum transverse deflection of the face sheets and the core crushing strain at mid-span of the panels. A limited set of simulations was carried out to find the optimum core density of a square honeycomb core sandwich panels under two shocks. The panels with a relative core density of 4%–5% are shown to have minimum face sheet deflection for the loading conditions considered here. This was consistent with the findings related to the sandwich panel response subjected to a single intense shock. Comparison of these results showed that optimized sandwich panels outperform solid plates under shock loading. An empirical method for prediction of the deflection and fracture of sandwich panels under two consecutive shocks – based on finding an effective peak over-pressure – was provided. Moreover, a limited number of simulations related to response and fracture of sandwich panels under multiple shocks with different material properties were performed to highlight the role of metal strength and ductility. In this set of simulations, square honeycomb sandwich panels made of four steels representing a relatively wide range of strength, strain hardening and ductility values were studied. For panels clamped at their edge, the observed failure mechanisms are core failure, top face failure and tearing at or close to the clamped edge. Failure diagrams for sandwich panels were constructed which reveal the fracture and failure mechanisms under various shock intensities for panels subjected to up to three consecutive shocks. The results complement previous studies on the behavior and fracture of these panels under high intensity dynamic loading and further highlights the potential of these panels for development of threat-resistant structural systems.     

A lattice deformation based model of metallic lattice sandwich plates subjected to impulsive loading

A lattice structure deformation mechanism based theoretical model is developed to predict the dynamic response of square lattice sandwich plates under impulsive loading. The analytical model is established on the basis of the three-stage framework proposed by Fleck and Deshpande (2004). In the first stage, the impulse transmitted from air shock loading to the sandwich plates by fluid-structure interaction is analytically calculated. The lattice core suffers non-uniform compression in the second stage due to the clamped boundary conditions. The structure deformation mechanism is introduced in the lattice core compression and the analytical nominal stress–strain curve of core compression accords well with previous experimental results. In the final stage, the sandwich plate is analyzed as a continuum plate with non-uniform thickness deduced by inconsistent deformation of the front and back sheets.The experiment results of square metallic sandwich plates with tetrahedral lattice core are presented and compared with analytical prediction to validate the theoretical model. Good agreements are found between the predicted and testing results for both the impulse transmitted to the sandwich plates and the maximum deflection of the back face sheet.     

The dynamic response of clamped rectangular Y-frame and corrugated core sandwich plates

The dynamic response of fully clamped, monolithic and sandwich plates of equal areal mass has been measured by loading rectangular plates over a central patch with metal foam projectiles. All plates are made from AISI 304 stainless steel, and the sandwich topologies comprise two identical face-sheets and either Y-frame or corrugated cores. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the plates as a function of projectile momentum. At low levels of projectile momentum both types of sandwich plate deflect less than monolithic plates of equal areal mass. However, at higher levels of projectile momentum, the sandwich plates tear while the monolithic plates remain intact. Three-dimensional finite element (FE) calculations adequately predict the measured responses, prior to the onset of tearing. These calculations also reveal that the accumulated plastic strains in the front face of the sandwich plates exceed those in the monolithic plates. These high plastic strains lead to failure of the front face sheets of the sandwich plates at lower values of projectile momentum than for the equivalent monolithic plates.     

A theoretical analysis of the dynamic response of metallic sandwich beam under impulsive loading

The objective of this paper is to analytically study the dynamic response of a fully clamped metallic sandwich beam under impulsive loading. The membrane factor method is employed to derive the solutions for large deflections and time responses of the sandwich beam, in which the interaction of bending and stretching is considered. Moreover, tighter ‘bounds’ of the solutions are obtained. It is shown that the present solutions are in good agreements with the previous finite element results and lie in the bounds of the solutions. It is clear that core strength and membrane force induced by large deflections have significant effects on the dynamic response of sandwich beam with increasing the transient deflections. The present method is efficient and simple for the dynamic response analysis of large deflections of metallic sandwich structures.     

The response of clamped sandwich plates with metallic foam cores to simulated blast loading

The dynamic responses of clamped circular monolithic and sandwich plates of equal areal mass have been measured by loading the plates at mid-span with metal foam projectiles. The sandwich plates comprise AISI 304 stainless steel face sheets and aluminium alloy metal foam cores. The resistance to shock loading is quantified by the permanent transverse deflection at mid-span of the plates as a function of projectile momentum. It is found that the sandwich plates have a higher shock resistance than monolithic plates of equal mass. Further, the shock resistance of the sandwich plates increases with increasing thickness of sandwich core. Finite element simulations of these experiments are in good agreement with the experimental measurements and demonstrate that the strain rate sensitivity of AISI 304 stainless steel plays a significant role in increasing the shock resistance of the monolithic and sandwich plates. Finally, the finite element simulations were employed to determine the pressure versus time history exerted by the foam projectiles on the plates. It was found that the pressure transient was reasonably independent of the dynamic impedance of the plate, suggesting that the metal foam projectile is a convenient experimental tool for ranking the shock resistance of competing structures.     

FREE VIBRATION ANALYSIS OF LATTICE SANDWICH BEAMS UNDER SEVERAL TYPICAL BOUNDARY CONDITIONS

Free vibration problems of lattice sandwich beams under several typical boundary conditions are investigated in the present paper. The lattice sandwich beam is transformed to an equivalent homogeneous three-layered sandwich beam. Unlike the traditional analytical model in which the rotation angles of the face sheets and the core are assumed the same, different rotation angles are considered in this paper to characterize the real response of sandwich beams. The analytical solutions of the natural frequencies for several typical boundary conditions are obtained. The effects of material properties and geometric parameters on the natural frequencies are also investigated.