Higher-order free vibrations of sandwich beams with a locally damaged core

The dynamical equations describing the free vibration of sandwich beams with a locally damaged core are derived using the higher-order theory approach. The nonlinear acceleration fields in the core are accounted for in the derivations, which is essential for the vibration analysis of the locally damaged sandwich beams. A local damage in the core, arbitrarily located along the length of the sandwich beam, is assumed to preclude the transition of stresses through the core between the undamaged parts of the beam. The damage is assumed to exist before the vibration starts and not to grow during oscillations. The numerical analysis based on the derived equations has been verified with the aid of the commercial finite element software ABAQUS. The numerical simulations reveal that a small local damage causes significant changes in the natural frequencies and corresponding vibration modes of the sandwich beams. An important practical consequence of the present work is that the vibration measurements can be successfully used as a nondestructive damage tool to assess local damages in sandwich beams.     

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.     

Low-energy impact response of composite and sandwich composite plates with piezoelectric sensory layers

An efficient model reduction based methodology is presented for predicting the global (impact force, plate deflection and electric potential) and through-thickness local (interfacial strains and stresses) dynamic response of pristine simply-supported cross-ply composite and sandwich composite plates with piezoelectric sensory layers subjected to low-energy impact. The through-thickness response of the laminate is modelled using coupled higher-order layerwise displacement-based piezoelectric laminate theories. Linearized contact laws are implemented for simulating the impactor–target interaction during impact. The stiffness, mass, piezoelectric and permittivity matrices of the plate are formulated from ply to structural level and reduced by applying a Guyan reduction technique to yield the structural system in state space. This reduction technique enables the formulation of a plate–impactor structural system of minimum size (1 term per vibration mode for composite plates – 2 terms for sandwich plates) and reduces computational cost, thus facilitating applicability for real-time impact and vibration control.     

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.     

曲壁蜂窝夹层板的振动特性研究

曲壁蜂窝具有负刚度特性,可以在大变形过程中吸收能量、抗冲击,并且在冲击过后可以自我恢复而不像传统蜂窝被压溃。本文将曲梁构成的负刚度蜂窝作为芯层,建立夹层板的动力学模型;推导出了曲壁负刚度蜂窝胞元的等效弹性参数,将其周期性排列为蜂窝芯,应用Reddy高阶剪切变形理论、Von-Karman大变形关系和Hamilton原理推导了负刚度蜂窝夹层板的非线性动力学方程;应用Navier法计算了四边简支边界条件下的固有频率。并利用有限元软件ABAQUS建立模型,计算固有频率,与理论计算结果进行比较,结果显示二者的计算结果具有较好的一致性,验证了芯层等效弹性参数及模型的有效性。探讨了在蜂窝胞元具有较高吸能情形下,夹层板在不同芯层厚度、不同芯厚比以及不同胞元曲壁厚度时的固有频率的变化特性。     

嵌锁式CFRP方形蜂窝夹芯梁低速冲击响应及失效机理

采用嵌锁组装工艺制备了碳纤维/树脂基复合材料方形蜂窝夹芯梁,实验研究了低速冲击载荷下简支和固支夹芯梁的动态响应及失效机理,获得了不同冲击速度下夹芯梁的失效模式,分析了其损伤演化过程和失效机理,探讨了冲击速度、边界条件、面板质量分布以及槽口方向等因素对夹芯梁破坏模式及承载能力的影响。研究结果表明,芯材长肋板槽口方向对夹芯梁的失效模式有较大影响,槽口向上的芯材跨中部分产生了挤压变形,而槽口向下的芯材跨中部分槽口在拉伸作用下出现了沿槽口开裂失效,继而引起面板脱粘和肋板断裂;同等质量下,较厚的上面板设计可以提高夹芯梁的抗冲击能力,冲击速度越大,夹芯梁的峰值载荷和承载能力越高;固支边界使得夹芯梁的后失效行为呈现出明显的强化效应,在夹芯梁跨中部分发生初始失效后出现了后继的固支端芯材和面板断裂失效模式。     

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。     

Theoretical analysis of the deformation of SMP sandwich beam in flexure

Shape memory polymers (SMPs) can have a large frozen strain but with a very small recovery stiffness in comparison with shape memory metals or ceramics. To provide more deployable stresses for the application of actuators, sandwich beams consisting of a SMP core and two thin metallic skins were considered. The packaging behaviors of two types of SMP sandwich beams, aluminum/SMP/aluminum and steel/SMP/steel, were discussed. Due to the high compliance of SMP core on packaging condition that the testing temperature is above the activation temperature of the material, buckling and post-buckling are the essential deformation mechanisms of SMP sandwich beams under bending. Theoretical solutions were derived in studying such non-linear behaviors, including the initiation of critical buckling, post-buckling response, and final failure modes. Systematic parameter’s analyses, e.g., buckling half-wavelength, amplitude, location of the neutral-strain surface in different packaging curvatures, were also presented.     

基于高阶变形理论的夹层板的弯曲与振动

考虑面板和夹芯的面内刚度和横向剪切刚度以及抗弯刚度,考虑了高阶剪切变形,根据横向剪应变分布情况给出横向剪切转角的位移函数,基于哈密尔顿原理,推导了基于高阶变形理论、适用于软、硬夹芯情况夹层板的基本方程。作为算例,以四边简支条件下的夹层板的弯曲与振动,在不同的面板与芯层的弹性模量比和厚度比下进行了计算,并与Reissner理论、Hoff理论以及邓宗白基于Reissner理论的修正模型的计算结果进行了对比。与前述理论与方法相比,本文方法考虑因素更为全面,对夹层板的适用范围更为广泛,计算结果更为精确。针对Nastran软件计算夹层板的振动问题,对其适用范围作了简要分析。     

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.     

Mechanical behavior of sandwich composite beams made of foams and functionally graded materials

We investigate sandwich composite beams using a direct approach which models slender bodies as deformable curves endowed with a certain microstructure. We derive general formulas for the effective stiffness coefficients of composite elastic beams made of several non-homogeneous materials. A special attention is given to sandwich beams with foam core, which are made of functionally graded or piecewise homogeneous materials. In the case of small deformations, the theoretical predictions are compared with experimental measurements for the three-point bending of sandwich beams, showing a very good agreement. For functionally graded sandwich columns we obtain the analytical solutions of bending, torsion and extension problems and compare them with numerical results computed by the finite element method.     

Higher-order dynamic response of composite sandwich panels with flexible core under simultaneous low-velocity impacts of multiple small masses

Small mass impactors, such as runway debris and hailstones may result in a wave controlled local response, which is essentially independent of boundary conditions. The higher-order impact model of sandwich beams presented by Mijia and Pizhong [Mijia, Y., Pizhong, Q., 2005. Higher-order impact modeling of sandwich structures with flexible core. International Journal of Solids and Structures 42 (10), 5460–5490] is developed and enhanced to impact analysis of sandwich panels with transversely flexible cores. Therefore, an improved fully dynamic higher-order impact theory is developed to analyze the low-velocity impact dynamic of a system which consists of a composite sandwich panel with transversely flexible core and multiple small impactors with small masses. Impacts are assumed to occur normally and simultaneously over the top face-sheet with arbitrary different masses and initial velocities of impactors. The contact forces between the panel and the impactors are treated as the internal forces of the system. First shear deformation theory (FSDT) is used for the face-sheets while three-dimensional elasticity is used for the soft core. The fully dynamic effects of the core layer and the face-sheets are considered in this study. Contact area can be varied with contact duration. The results in multiple mass impacts over sandwich panels that are hitherto not reported in the literature are presented based on proposed improved higher-order sandwich plate theory (IHSAPT). Finally, for the case study of the single mass impact, the numerical results of the analysis have been compared either with the available experimental results or with some theoretical results. As no literature could be found on the impact of multiple impactors over sandwich panels, the present formulation is validated indirectly by comparing the response of two cases of double small masses and single small mass impacts. Also, in order to demonstrate the applicability of the validation, the analytical relation of minimum distance between two impactors is derived based on Olsson’s wave control principle in this paper.     

Response and Damage Tolerance of Composite Sandwich Structures under Low Velocity Impact

The deformation and failure response of composite sandwich beams and panels under low velocity impact was reviewed and discussed. Sandwich facesheet materials discussed are unidirectional and woven carbon/epoxy, and woven glass/vinylester composite laminates; sandwich core materials investigated include four types of closed cell PVC foams of various densities, and balsa wood. Sandwich beams were tested in an instrumented drop tower system under various energy levels, where load and strain histories and failure modes were recorded for the various types of beams. Peak loads predicted by spring-mass and energy balance models were in satisfactory agreement with experimental measurements. Failure patterns depend strongly on the impact energy levels and core properties. Failure modes observed include core indentation/cracking, facesheet buckling, delamination within the facesheet, and debonding between the facesheet and core. In the case of sandwich panels, it was shown that static and impact loads of the same magnitude produce very similar far-field deformations. The induced damage is localized and is lower for impact loading than for an equivalent static loading. The load history, predicted by a model based on the sinusoidal shape of the impact load pulse, was in agreement with experimental results. A finite element model was implemented to capture the full response of the panel indentation. The investigation of post impact behavior of sandwich structures shows that, although impact damage may not be readily visible, its effects on the residual mechanical properties of the structure can be quite detrimental.     

Low-velocity impact response of sandwich beams with functionally graded core

The problem of low-speed impact of a one-dimensional sandwich panel by a rigid cylindrical projectile is considered. The core of the sandwich panel is functionally graded such that the density, and hence its stiffness, vary through the thickness. The problem is a combination of static contact problem and dynamic response of the sandwich panel obtained via a simple nonlinear spring-mass model (quasi-static approximation). The variation of core Young’s modulus is represented by a polynomial in the thickness coordinate, but the Poisson’s ratio is kept constant. The two-dimensional elasticity equations for the plane sandwich structure are solved using a combination of Fourier series and Galerkin method. The contact problem is solved using the assumed contact stress distribution method. For the impact problem we used a simple dynamic model based on quasi-static behavior of the panel—the sandwich beam was modeled as a combination of two springs, a linear spring to account for the global deflection and a nonlinear spring to represent the local indentation effects. Results indicate that the contact stiffness of the beam with graded core increases causing the contact stresses and other stress components in the vicinity of contact to increase. However, the values of maximum strains corresponding to the maximum impact load are reduced considerably due to grading of the core properties. For a better comparison, the thickness of the functionally graded cores was chosen such that the flexural stiffness was equal to that of a beam with homogeneous core. The results indicate that functionally graded cores can be used effectively to mitigate or completely prevent impact damage in sandwich composites.     

High-order layerwise mechanics and finite element for the damped dynamic characteristics of sandwich composite beams

A high-order discrete-layer theory and a finite element are presented for predicting the damping of laminated composite sandwich beams. The new layerwise laminate theory involves quadratic and cubic terms for approximation of the in-plane displacement in each discrete layer, while interlaminar shear stress continuity is imposed through the thickness. Integrated damping mechanics are formulated and both laminate and structural stiffness, mass and damping matrices are formed. A finite element method and a beam element are further developed for predicting the free vibration response, including modal frequencies, modal loss factors and through-thickness mode shapes. Numerical results and evaluations of the present model are shown. Modal frequencies and damping of sandwich composite beams are measured and correlated with predicted values. Finally, parametric studies illustrate the effect of core thickness and face lamination on modal damping and frequency values.     

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.     

Experimental and Numerical Investigation of Free Vibrations of Composite Sandwich Beams with Curvature and Debonds

In this paper experimental and numerical results concerning the dynamic response of composite sandwich beams with curvature and debonds are reported. Sandwich beams made of carbon/epoxy face sheets and polyurethane foam core material were manufactured with four different radii of curvature and debonds between the top and bottom interface of face sheet and foam core. Dynamic response was obtained using the impulse frequency response technique under clamped-clamped boundary condition. Experimental results were compared with numerical finite element model results. A combined experimental and numerical FE approach was used to determine the material properties of the skin and foam core materials based on modal vibration and static flexure tests. Results indicate that the fundamental frequency increases with increasing curvature angle, however, for higher frequencies; the natural frequencies are not significantly affected. Also, it is found that face/core debond causes reduction of the natural frequencies due to stiffness degradation.     

多孔金属及其夹芯结构力学性能的研究进展

高孔隙率多孔金属及其夹芯复合结构是一种物理功能与结构一体化的新型、轻质高强材料/结构,具有高比强度、高比刚度和优良的吸能和缓冲性能等多种功能,引起了学术界和工程界众多研究者的极大关注. 本文概述了轻质多孔金属及其夹芯结构的制备方法、多功能特性及其应用,介绍了多孔金属夹芯结构元件(梁、板、壳)遭受准静态和动态冲击载荷下的理论、实验和模拟方面的国内外研究现状,分析和讨论了多孔金属及其夹芯结构力学行为研究中的研究手段和基本问题,重点关注了多孔金属夹芯结构的变形/失效、动态响应和能量吸收.     

曲壁蜂窝夹层悬臂板的振动特性研究

蜂窝结构作为一种多孔材料具有轻质、高强度、高刚度的优点, 兼具隔声降噪、隔热等优良性能, 被广泛应用于交通运输、航空航天等领域. 传统直壁蜂窝在受力后容易出现应力集中的问题, 这将导致蜂窝夹层产生裂纹破坏, 缩短夹层板的使用寿命. 针对此问题本文设计了一种以圆弧曲壁蜂窝作为芯层的蜂窝夹层板, 基于单位载荷法推导了蜂窝芯的等效参数, 建立曲壁蜂窝夹层板的动力学模型, 利用Chebyshev-Ritz方法求解悬臂边界下曲壁蜂窝夹层板的固有频率, 并用有限元方法进行对比验证, 发现前5阶固有频率的误差均在5%以内, 每阶固有频率对应的振型一致. 通过3D打印聚乳酸(PLA)制备了曲壁蜂窝夹层板, 使用万能试验机对PLA拉伸试件进行准静态拉伸测定了打印材料的杨氏模量, 搭建振动试验平台对制备的曲壁蜂窝夹层板进行正弦扫频试验、定频谐波驻留试验和冲击试验. 对比发现3D打印模型振动试验获得的前5阶固有频率与理论模型和有限元模型的计算结果三者一致, 试验发现曲壁蜂窝芯在特定频段内具有一定的抗冲击性能. 研究结果将为曲壁蜂窝在振动和隔振方面的应用提供理论支持.      

Strength and stiffness of sandwich beams in bending

This investigation is concerned with the experimental versus analytical correlation of the mechanical properties of sandwich-beam specimens. Such sandwich structures are commonly employed in the aircraft industry. Four-point and three-point load tests were conducted on a large number of sandwich-beam specimens, fabricated by using fiber-glass reinforced plastics (both unidirectional and woven-glass cloth) and DTD 685 aluminum alloy for the facings with aluminum honeycomb core and polyurethane foam cores and the indigenously available Araldite as the bonding medium between the core and the facings.The flexural stiffness of the composite sandwich specimens used in this investigation compared favorably with theoretical predictions. The shear stiffness was found to be about 55 percent and 45 percent of the theoretically predicted values for FRP (fiberglass-reinforced-plastic) cloth and FRP unidirectional laminates with aluminum honeycomb core sandwich, respectively. The failure load as determined by experiments was less than the theoretically predicted safe load. There was a loss of strength as well as a steep decrease in the failure load in the case of low density foam core.It was concluded that FRP facing plates with aluminum honeycomb core sandwich structure may be preferred to similar aluminum-alloy facing sandwich construction if high flexural stiffness and shear stiffness properties are required at less cost and weight. Indigenously available Araldite was quite satisfactory for bonding the core to the facings.This investigation has confirmed the importance of experiments in the field of sandwich structures which can effectively replace other conventional uneconomical structural or machine members which are currently in use.