Nonlinear Transient Response of Functionally Graded Material Sandwich Doubly Curved Shallow Shell Using New Displacement Field

In this paper, the nonlinear transient dynamic response of functionally graded material(FGM) sandwich doubly curved shell with homogenous isotropic material core and functionally graded face sheet is analyzed using a new displacement field on the basis of Reddy's third-order shear theory for the first time. The equivalent material properties for the FGM face sheet are assumed to obey the rule of simple power law function in the thickness direction.Based on Reddy'stheory of higher shear deformation, a new displacement field is developed by introducing the secant function into transverse displacement. Four coupled nonlinear differential equations are obtained by applying Hamilton's principle and Galerkin method. It is assumed that the FGM sandwich doubly curved shell is subjected to step loading, air-blast loading,triangular loading, and sinusoidal loading, respectively. On the basis of double-precision variablecoefficient ordinary differential equation solver, a new program code in FORTRAN software is developed to solve the nonlinear transient dynamics of the system. The influences of core thickness, volume fraction, core-to-face sheet thickness ratio, width-to-thickness ratio and blast type on the transient response of the shell are discussed in detail through numerical simulation.     

ELASTIC-PLASTIC DYNAMIC RESPONSE OF FULLY BACKED SANDWICH PLATES UNDER LOCALIZED IMPULSIVE LOADING

An analytical model is developed to assess the elastic-plastic dynamic response of fully backed sandwich plates under localized impulse load.The core is modeled as an elastic-perfectly plastic foundation.The top face sheet is treated as an individual plate resting on the foundation.The elastic-plastic analysis for the top face sheet is based on a minimum principle in dynamic plasticity associated with the finite difference technique.The effects of spatial and temporal distributions of the impulsive loading on the dynamic response of sandwich plates are discussed.The model can be used to predict the impulse-induced local effect on fully backed sandwich plates.     

Dynamic response of composite sandwich plates subjected to time-dependent pressure pulses

The dynamic response of orthotropic sandwich composite plates impacted by time-dependent external blast pulses is studied by use of numerical techniques. The theory is based on classical sandwich plate theory including the large deformation effects, such as geometric non-linearities, in-plane stiffness and inertias, and shear deformation. The equations of motion for the plate are derived by the use of the virtual work principle. Approximate solutions are assumed for the space domain and substituted into the equations of motion. Then the Galerkin Method is used to obtain the non-linear differential equations in the time domain. The finite difference method is applied to solve the system of coupled non-linear equations. The results of theoretical analyses are obtained and compared with ANSYS results. Effects of the face sheet number, as well as those related to the ply-thickness, core thickness, geometrical non-linearities, and of the aspect ratio are investigated. Detailed analyses of the influence of different type of pressure pulses on dynamic response are carried out.     

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.     

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.     

撞击载荷下泡沫铝夹层板的动力响应

应用泡沫金属子弹撞击加载的方式研究了固支方形夹层板和等质量实体板的动力响应,分别应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历史,给出了夹层板的变形与失效模式,研究了子弹冲量、面板厚度、泡沫芯层厚度及芯层密度对夹层板抗撞击性能的影响。结果表明,后面板中心点挠度最大,周边最小,整体变形为穹形,且伴有花瓣形的变形。参数研究表明,通过增加面板厚度或芯层厚度均能有效控制后面板的挠度,改善夹层板的能量吸收能力,结构响应对子弹冲量和芯层密度比较敏感。实验结果对多孔金属夹层结构的优化设计具有一定的参考价值。 更多还原     

Non-linear response of curved sandwich panels – extended high-order approach

The geometrical non-linear behavior a curved sandwich panel with a stiff or compliant core when subjected to a pressure load using the Extended High-Order Sandwich Panel theory (EHSAPT), is presented. The formulation follows the EHSAPT procedure where the in-plane. i.e circumferential rigidity of the core is considered and the distribution of the displacements through the depth of the core are presumed. These displacement distributions are the closed-form solutions of the 2D governing equations of the curved core without circumferential rigidity that appear in the HSAPT curved sandwich panel model. The mathematical formulation includes the field equations along with the appropriate boundary and continuity conditions that take into account the high-order stress resultants in the core due to the presumed distributions. Finally a numerical study is conducted for a panel loaded by a distributed pressure at the upper face sheet. It reveals that the post-buckling response of a curved sandwich panels is associated with shallow to deep wrinkling deformations of the upper face sheet in the case of a simply-supported panel or a general non-linear pattern without wrinkles in the case of pinned supports with a short span. In both cases a stable post-buckling response is observed similar to that of a plate one.     

Local slamming impact of sandwich composite hulls

We develop a hydroelastic model based on a {3, 2}-order sandwich composite panel theory and Wagner’s water impact theory for investigating the fluid–structure interaction during the slamming process. The sandwich panel theory incorporates the transverse shear and the transverse normal deformations of the core, while the face sheets are modeled with the Kirchhoff plate theory. The structural model has been validated with the general purpose finite element code ABAQUS^®. The hydrodynamic model, based on Wagner’s theory, considers hull’s elastic deformations. A numerical procedure to solve the nonlinear system of governing equations, from which both the fluid’s and the structure’s deformations can be simultaneously computed, has been developed and verified. The hydroelastic effect on hull’s deformations and the unsteady slamming load have been delineated. This work advances the state of the art of analyzing hydroelastic deformations of composite hulls subjected to slamming impact.     

Dynamic geometrically nonlinear analysis of transversely compressible sandwich plates

In order to construct a plate theory for a thick transversely compressible sandwich plate with composite laminated face sheets, the authors make simplifying assumptions regarding distribution of transverse strain components in the thickness direction. The in-plane stresses and σ_yy (Fig. 1) are computed from the constitutive equations, and the improved values of transverse stress components and σ_zz need to be computed by integration of pointwise equations of motion in a post-process stage of the finite element analysis. The improved values of the transverse strains can also be computed in the post-process stage by substituting the improved transverse stresses into the constitutive relations. A problem of cylindrical bending of a simply supported plate under a uniform load on the upper surface is considered, and comparison is made between the displacements, the in-plane stress and the improved transverse stresses (obtained by integration of the pointwise equations of motion), computed from the plate theory, with the corresponding values of exact elasticity solutions. In this comparison, a good agreement of both solutions is achieved. In the finite element analysis of sandwich plates in cylindrical bending with small thickness-to-length ratios, the shear locking phenomenon does not occur. The model of a sandwich plate in cylindrical bending, presented in this paper, has a wider range of applicability than the models presented in literature so far: it can be applied to the sandwich plates with a wide range of ratios of thickness to the in-plane dimensions, with both thin and thick face sheets (as compared to the thickness of the core) and to the sandwich plates with both transversely rigid and transversely compressible face sheets and cores.     

Analysis and active control of nonlinear vibration of composite lattice sandwich plates

This paper is devoted to investigate the nonlinear vibration characteristics and active control of composite lattice sandwich plates using piezoelectric actuator and sensor. Three types of the sandwich plates with pyramidal, tetrahedral and Kagome cores are considered. In the structural modeling, the von Kármán large deflection theory is applied to establish the strain–displacement relations. The nonlinear equations of motion of the structures are derived by Hamilton’s principle with the assumed mode method. The nonlinear free and forced vibration responses of the lattice sandwich plates are calculated. The velocity feedback control (VFC) and H_∞ control methods are applied to design the controller. The nonlinear vibration responses of the sandwich plates with pyramidal, tetrahedral and Kagome cores are compared. The influences of the ply angle of the laminated face sheets, the thicknesses of the lattice core and face sheets and the excitation amplitude on the nonlinear vibration behaviors of the sandwich plates are investigated. The correctness of the H_∞ control algorithm is verified by comparing with the experiment results reported in the literature. The controlled nonlinear vibration response of the sandwich plate is computed and compared with that of the uncontrolled structural system. Numerical results indicate that the VFC and H_∞ control methods can effectively suppress the large amplitude vibration of the composite lattice sandwich plates.

     

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.     

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.     

Critical velocity of sandwich cylindrical shell under moving internal pressure

Critical velocity of an infinite long sandwich shell under moving internal pressure is studied using the sandwich shell theory and elastodynamics theory. Propagation of axisymmetric free harmonic waves in the sandwich shell is studied using the sandwich shell theory by considering compressibility and transverse shear deformation of the core, and transverse shear deformation of face sheets. Based on the elastodynamics theory, displacement components expanded by Legendre polynomials, and position-dependent elastic constants and densities are introduced into the equations of motion. Critical velocity is the minimum phase velocity on the desperation relation curve obtained by using the two methods. Numerical examples and the finite element （FE） simulations are presented. The results show that the two critical velocities agree well with each other, and two desperation relation curves agree well with each other when the wave number k is relatively small. However, two limit phase velocities approach to the shear wave velocities of the face sheet and the core respectively when k limits to infinite. The two methods are efficient in the investigation of wave propagation in a sandwich cylindrical shell when k is relatively small. The critical velocity predicted in the FE simulations agrees with theoretical prediction.     

双层夹芯复合材料结构横向冲击响应实验

采用玻璃纤维增强环氧树脂复合材料层合板作为内、外面板,以PVC泡沫作为芯材,构造了双层夹芯复合材料结构。采用落锤冲击实验,得到了冲击过程的撞击力历史;研究了在不同的冲击能量下,双层夹芯结构的冲击响应及内面板位置对双层夹芯结构冲击响应的影响。实验结果表明,内面板的引入及内面板的位置显著影响双层夹芯结构的撞击力历史,根据该撞击力历史可以优化设计出抗冲击性能优异的新型双层夹芯复合材料结构。     

PVC夹芯板在冲击载荷下的动态响应与失效模式

通过开展对泡沫金属子弹撞击加载聚氯乙烯(polyvinyl chloride, PVC)夹芯板的实验,结合三维数字图像相关性(three dimensional digital image correlation, DIC-3D)技术,研究固支夹芯板在撞击加载条件下的动态响应,获得夹芯板受撞击及响应的变形过程,并结合图像分别分析夹芯板整体及三层结构的变形和失效模式;研究子弹冲量与背板最终变形之间的关系和相似冲量下等面密度不同芯层密度的夹芯结构的抗撞击性能。结果表明:夹芯板的破坏和失效主要集中在泡沫金属子弹直接作用区域,背板挠度由中间向固定端逐渐减小,子弹冲量与背板变形近似成线性关系。在等质量的条件下,降低芯层密度、增加芯层厚度可以有效降低背板的变形,实验结果对聚合物夹芯结构的工程优化设计具有一定的参考意义。     

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.     

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.     

Sandwich column buckling – A hyperelastic formulation

The macro-buckling equations for a sandwich column are developed. A layer-wise Timoshenko beam displacement approximation is assumed. The constitutive relationships and equilibrium equations for the core and face sheets are derived using a consistent hyperelastic neo-Hookean formulation. The derivations in this paper are consistent with that of Haringx’s and Reissner’s proposal for beam actions. The buckling formulation includes the axial deformation prior to buckling and the transverse shear deformation of the core and face sheets. The buckling equations derived agree with the equation of [Allen, H.G., 1969. Analysis and Design of Structural Sandwich Panels, Pergamon, Oxford] for thick faces but are also applicable to any ratio of face sheet to core thickness and material properties. The formulation is compared to experimental results for sandwich columns and shows good comparison except for very short columns. The formulation is also compared to the buckling experimental results for short rubber rods and also compared well. The formulation does not predict a shear buckling mode.     

BENDING OF ORTHOTROPIC SANDWICH PLATES WITH A FUNCTIONALLY GRADED CORE SUBJECTED TO DISTRIBUTED LOADINGS

Based on the Reissner assumptions, this paper is concerned with the bending analysis of simply supported sandwich plates with functionally graded core and orthotropic face sheets subjected to transverse distributed loadings. First, the expressions of the displacements, stresses and internal forces of the sandwich plate are presented according to the constitutive relations and stress states of the core and face sheets. Then, the solutions of bending equilibrium equations are derived by expanding the deflection w, transverse shearing forces Q_x and Q_y with double trigonometric series that satisfy the simply supported boundary conditions. Finally, the proposed solution is validated by comparing the results with available elasticity solutions for a square sandwich plate with an isotropic core and finite element simulations for one with functionally graded core. The Young's modulus of the functionally graded core is assumed to be graded by a power law distribution of volume fractions of the constituents, and the Poisson's ratio is held constant. And the effects of the core's top-bottom Young's modulus ratio ? and volume fraction exponent n₀ on the variation of the displacements of the functionally graded sandwich plate are also examined.     

热环境下功能梯度夹层双曲壳三维自由振动分析

功能梯度夹层双曲壳结构广泛应用在航空航天、海洋工程等领域中,对于该类结构的动力学特性研究非常重要。本文以热环境下功能梯度夹层双曲壳为研究对象,在三阶剪切变形理论的基础上,考虑横向拉伸作用的影响提出了一种新的位移场,假设材料的物性参数与温度有关,且沿厚度方向表示为幂律函数。利用Hamilton原理得到简支边界条件下功能梯度夹层双曲壳三维振动系统动力学方程,利用Navier法求得两种不同夹层类型的系统固有频率。研究了几何物理参数和温度场对功能梯度夹层双曲壳自由振动固有频率的影响。