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.     

One-dimensional analytical model for the response of elastic coatings to water blast

In previous studies, the response of sandwich structures to water blast is solved by envisaging the plate adjacent to water as a rigid one, while the effects of the elasticity in the longitudinal direction of the plate are rarely studied. In this paper, a monolithic elastic coating with varying stiffness and thickness is investigated by a one-dimensional analytical approach, based on linear wave motion theory, to reveal the elastic effect of the plate on the incident wave. One side of the coating is loaded by a planar shock wave; on the opposite side, rigid boundary or air-backed boundary is imposed. The fluid–structure interaction (FSI), cavitation phenomenon and large deformation of the coating are taken into account. In particular, the initiation and evolution of cavitation, including the propagations of breaking fronts and closing fronts, as well as the pressure histories of radiated wave by the closing front, are examined. The analytical solution has been compared with finite element (FE) predictions. The results are found to be in excellent agreement for the propagation of breaking front and closing front, as well as the pressure and particle velocity histories at the wet face before the cavitation reaches the wet face. However, when the wet face cavitates, the predictions provided by the analytical method are less accurate and the analytically-computed particle velocity can only be compared in an average sense with the FE predictions. For air-backed case, Taylor׳s model prior to cavitation becomes a trivial case of the analytical model and the comparison also indicates the validity of the analytical model. The validated analytical model is used to determine the dependence of the peak pressure at the wet face and the impulse transmitted to the coating on the coating properties, including the wave impedance and thickness.     

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.     

Point-wise impulse (blast) response of a composite sandwich plate including core compressibility effects

This work analyzes the nonlinear impulse response of a composite sandwich plate exposed to a sudden point-wise transverse loading on the top face sheet. The nonlinearity arising from the core compressibility in the thickness direction is modeled and incorporated into the constitutive relations explicitly. As such, one can have a deep insight regarding the stress, strain and displacement profiles into the sandwich plate. The sandwich plate is assumed to be perfectly bonded at the face sheet/core interfaces. The equations of motion are formulated using Hamilton’s principle. The simply supported case is used to illustrate the procedure for solving the nonlinear equations. Numerical results are presented to demonstrate the response in terms of the transverse deformation and stresses in the composite sandwich plate. The effects of the variation of the geometrical parameters of the structure on the blast impulse response are also studied. Some conclusions are suggested regarding the associated optimal design of sandwich plates.     

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。     

Metal sandwich plates subject to intense air shocks

Recent results on fluid–structure interaction for plates subject to high intensity air shocks are employed to assess the performance of all-metal sandwich plates compared to monolithic solid plates of the same material and mass per area. For a planar shock wave striking the plate, the new results enable the structural analysis to be decoupled from an analysis of shock propagation in the air. The study complements prior work on the role of fluid–structure interaction in the design and assessment of sandwich plates subject to water shocks. Square honeycomb and folded plate core topologies are considered. Fluid–structure interaction enhances the performance of sandwich plates relative to solid plates under intense air shocks, but not as significantly as for water blasts. The paper investigates two methods for applying the loading to the sandwich plate—responses are contrasted for loads applied as a time-dependent pressure history versus imposition of an initial velocity.     

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.     

Dynamics of cavitational demolition in the interaction of a shock wave with a free surface

The problem of cavitation in the tension wave associated with the reflection of a shock wave from a free fluid surface is considered. A method of calculating the cavitation zone dynamics which makes it possible to determine the structure of the cavitation front, including for large space scales, is developed. A procedure for determining the dispersity of the fragment-drops of dispersed fluid, which takes into account the initial size distribution of the cavitation nuclei and the parameters of the incident shock wave, is proposed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.6, pp. 73–80, November–December, 1992.     

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.     

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.     

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 comprehensive analysis of functionally graded sandwich plates: Part 1—Deflection and stresses

A two-dimensional solution is presented for bending analysis of simply supported functionally graded ceramic–metal sandwich plates. The sandwich plate faces are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity and Poisson’s ratio of the faces are assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The core layer is still homogeneous and made of an isotropic ceramic material. Several kinds of sandwich plates are used taking into account the symmetry of the plate and the thickness of each layer. We derive field equations for functionally graded sandwich plates whose deformations are governed by either the shear deformation theories or the classical theory. Displacement functions that identically satisfy boundary conditions are used to reduce the governing equations to a set of coupled ordinary differential equations with variable coefficients. Numerical results of the sinusoidal, third-order, first-order and classical theories are presented to show the effect of material distribution on the deflections and stresses.     

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.     

夹芯圆柱壳在水下爆炸载荷作用下的抗冲击特性

采用声固耦合方法对夹芯圆柱壳和等质量的普通圆柱壳在爆炸载荷作用下的应变、速度和加速度进行有限元计算。结果表明:夹芯防护层对爆炸冲击波可起到较好的衰减作用,即通过芯层的塑性变形,耗散了冲击过程中产生的大部分能量,对里面的圆柱壳体起到较好的保护作用,由于夹芯防护层的存在,与等质量的普通圆柱壳相比,夹芯圆柱壳能够承受更强的爆炸冲击波,降低结构的整体变形。     

Shock structure in bubbly liquids: comparison of direct numerical simulations and model equations

Shock wave structure in a bubbly mixture composed of a cluster of gas bubbles in a quiescent liquid with initial void fractions around 10% inside a 3D rectangular domain excited by a sudden increase in the pressure at one boundary is investigated using the front tracking/finite volume method. The effects of bubble/bubble interactions and bubble deformations are, therefore, investigated for further modeling. The liquid is taken to be incompressible while the bubbles are assumed to be compressible. The gas pressure inside the bubbles is taken uniform and is assumed to vary isothermally. Results obtained for the pressure distribution at different locations along the direction of propagation show the characteristics of one-dimensional unsteady shock propagation evolving towards steady-state. The steady-state shock structures obtained by the present direct numerical simulations, which show a transition from A-type to C-type steady-state shock structures, are compared with those obtained by the classical Rayleigh–Plesset equation and by a modified Rayleigh–Plesset equation accounting for bubble/bubble interactions in the mean-field theory.      

Dynamics of a “collective” bubble in a magma melt flow behind the decompression wave front

Specific features of the dynamics of the wave field structure and growth of a “collective” bubble behind the decompression wave front in the “Lagrangian” section of the formed cavitation zone are numerically analyzed. Two cases are considered: with no diffusion of the dissolved gas from the melt to cavitation nuclei and with the diffusion flux providing an increase in the gas mass in the bubbles. In the first case, it is shown that an almost smooth decompression wave front approximately 100 m wide is formed, with minor perturbations that appear when the front of saturation of the cavitation zone with nuclei is passed. In the case of the diffusion process, the melt state behind the saturation front is principally different: jumps in mass velocity and viscosity are observed in the vicinity of the free surface, and the pressure in the “collective” cavitation bubble remains unchanged for a sufficiently long time interval, despite the bubble growth and intense diffusion of the gas from the melt. It is assumed that the diffusion process (and, therefore, viscosity) actually become factors determining the dynamics of growth of cavitation bubbles beginning from this time interval. A pressure jump is demonstrated to form near the free surface.     

An analytic model for the response to water blast of unsupported metallic sandwich panels

A prior assessment of the response of a metallic sandwich panels to water blast has identified soft and strong core responses and outlined the advantages of softness. Ensuing analysis has provided mechanism maps that distinguish these responses. The present article extends these assessments by developing an analytic model for the wet face response, inclusive of fluid/structure interaction, that can be used for a wide range of core topologies. The model addresses cavitation and incorporates the momentum of reconstituted water attached to the wet face. It assumes a transient dynamic strength of the core associated with dynamic buckling. The model includes coefficients that have been independently characterized using numerical simulations. The fidelity of the analytic model has also been assessed using simulations. The results reveal that analytic predictions of the wet face velocities are quite accurate for most of the soft cores examined. The implication is that the models may be used as reliable input to panel-level simulations for predicting such metrics as the reaction forces and the displacements. Discrepancies arise for strong cores with relatively large push back stress, and for systems with very thin wet faces, suggesting that embellishments are required for panels incorporating such features.     

Uniaxial crushing of sandwich plates under air blast: Influence of mass distribution

Motivated by recent efforts to mitigate blast loading using energy-absorbing materials, this paper uses analytical and computational modeling to investigate the influence of mass distribution on the uniaxial crushing of cellular sandwich plates under air blast loading. In the analytical model, the cellular core is represented using a rigid, perfectly-plastic, locking idealization, as in previous studies, and the front and back faces are modeled as rigid, with pressure loading applied to the front face and the back-face unrestrained. This model is also applicable to the crushing of cellular media in “blast pendulum” experiments. Fluid–structure interaction effects are treated using a recent result that accounts for nonlinear compressibility effects for intense air blasts. Predictions of the analytical model show excellent agreement with explicit finite element computations, and the model is used to investigate the response of the system for all possible distributions of mass between the front and back faces and the cellular core. Increasing the mass fraction in the front face is found to increase the impulse required for complete crushing of the cellular core but also to produce undesirable increases in back-face accelerations. Optimal mass distributions for mitigating shock transmission through the sandwich plate are investigated by maximizing the impulse capacity while limiting the back-face accelerations to a specified level.     

一种鲁棒的HLLC格式及其稳定性分析

精确捕捉接触波和剪切波的Godunov型数值方法,如流行的HLLC格式,在模拟高超声速流动问题时会出现激波异常现象。对HLLC格式进行稳定性分析发现,流体主流方向的扰动都能有效衰减,但是横向的密度与剪切速度的扰动不会衰减。具有特殊对称性的二维Sedov爆轰波问题证明了横向通量和不稳定现象之间的密切联系。利用压力比和马赫数来探测数值激波层亚声速区的横向网格界面,并且在该界面的数值通量上增加熵波粘性和剪切波粘性来构造一种激波稳定的HLLC格式。分析表明,在熵波粘性和剪切波粘性的作用下,横向的所有扰动都会衰减。一系列数值测试证明了新格式不仅可以成功地抑制各类激波异常现象,还保留了原HLLC格式低耗散性的优点。     

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.