Dynamic response of clamped sandwich beams: analytical modeling

An improved analytical model is developed to predict the dynamic response of clamped lightweight sandwich beams with cellular cores subjected to shock loading over the entire span.The clamped face sheets are simplified as a single-degree-of-freedom(SDOF) system, and the core is idealized using the rigid-perfectly-plastic-locking(RPPL) model. Reflection of incident shock wave is considered by incorporating the bending/stretching resistance of the front face sheet and compaction of the core. The model is validated with existing analytical predictions and FE simulation results, with good agreement achieved. Compared with existing analytical models, the proposed model exhibits superiority in two aspects: the deformation resistance of front face sheet during shock wave reflection is taken into account; the effect of pulse shape is considered. The practical application range of the proposed model is therefore wider.     

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).     

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.     

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.     

The response of circular composite plates to underwater blast: Experiments and modelling

We present a new experimental technique to allow laboratory-scale observation of underwater blast loading on circular plates, including dynamic deformation and failure of the plates as well as the sequence of cavitation events in water. The apparatus is used to measure and compare the responses of a quasi-isotropic glass/vinylester composite and of a woven carbon/epoxy plate. Dynamic explicit FE simulations are conducted and their predictions are found in good agreement with experiments. Measurements and FE predictions are used to validate a recently developed theoretical model for the response of elastic orthotropic plates to underwater blast.     

破碎带波浪的数值模拟

基于一组色散关系得到改进的完全非线性Boussinesq方程建立了一个波浪模型可以模拟近岸水域的波浪变浅、破碎以及在海滩上的爬高等多种变形。波浪破碎引起的能量衰减是在动量方程中引入一个在空间和时间上都只作用于波前的涡粘项来模拟。动海岸线边界用窄缝法处理。波浪爬高用非线性浅水方程推导的非破碎波浪在斜坡上爬高的解析解来验证。本模型还模拟了波浪在斜坡上不同类型的破碎变形过程,并将其波高和平均水位的沿程变化和物理模型实验的结果比较,两者符合良好。     

An analytic model for the water blast response of one-dimensional marine structure coated with elastic foam

The water blast response of one-dimensional marine structures coated with elastic foam is theoretically investigated. The foam coating is modeled by a group of concentrated masses separated by parallel massless nonlinear spring and damper, simulating the micro inertia, stiffness, and rate-dependent effects exhibited by common cellular materials. An analytic model for the wet face response, inclusive of fluid-structure interaction, is developed. The model addresses cavitation and takes into account the momentum of reconstituted water attached to the wet face. The first-order double asymptotic approximation method is used to compute the pressure during the spring-back and succeeding phase. The fidelity of the analytic model has been assessed by the finite element analysis.     

应用阵面追踪法对散心爆轰波传播的数值模拟

考虑到爆轰波阵面曲率及化学反应区宽度的作用 ,由修正Hugoniot关系式解析求解了波后状态 ,应用阵面追踪法 (FTM)数值模拟了曲面散心爆轰波的传播。计算给出的散心爆轰波走时及波阵面上物理量都达到了波阵面曲率的一阶精度。     

Macro-mechanical modeling of blast-wave mitigation in foams. Part II: reliability of pressure measurements

A phenomenological study of the process occurring when a plane shock wave reflected off an aqueous foam column filling the test section of a vertical shock tube has been undertaken. The experiments were conducted with initial shock wave Mach numbers in the range $1.25\le {M}_\mathrm{s} \le 1.7$ and foam column heights in the range 100–450 mm. Miniature piezotrone circuit electronic pressure transducers were used to record the pressure histories upstream and alongside the foam column. The aim of these experiments was to find a simple way to eliminate a spatial averaging as an artifact of the pressure history recorded by the side-on transducer. For this purpose, we discuss first the common behaviors of the pressure traces in extended time scales. These observations evidently quantify the low frequency variations of the pressure field within the different flow domains of the shock tube. Thereafter, we focus on the fronts of the pressure signals, which, in turn, characterize the high-frequency response of the foam column to the shock wave impact. Since the front shape and the amplitude of the pressure signal most likely play a significant role in the foam destruction, phase changes and/or other physical factors, such as high capacity, viscosity, etc., the common practice of the data processing is revised and discussed in detail. Generally, side-on pressure measurements must be used with great caution when performed in wet aqueous foams, because the low sound speed is especially prone to this effect. Since the spatial averaged recorded pressure signals do not reproduce well the real behaviors of the pressure rise, the recorded shape of the shock wave front in the foam appears much thicker. It is also found that when a thin liquid film wet the sensing membrane, the transducer sensitivity was changed. As a result, the pressure recorded in the foam could exceed the real amplitude of the post-shock wave flow. A simple procedure, which allows correcting this imperfection, is discussed in detail.     

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.     

高速杆式弹体侵彻下蓄液结构载荷特性的有限元分析

为探讨高速弹体侵彻下蓄液结构的防护方法,采用瞬态非线性有限元,研究了高速杆式弹体侵彻下蓄液结构承受的冲击载荷特性,分析了冲击载荷的作用过程、前后板承受的载荷强度及其弹体初速度和水域尺度的影响。结果表明:弹体在蓄液结构中的初始开坑作用,将形成入射冲击波,其压力峰值极高,但作用时间短,并将在液体内产生多次反射;弹体在液体中的侵彻,将产生空化,并形成峰值小、作用时间长的空化压力载荷;后板对液体流的阻碍作用将形成出口局部高压;入射冲击波和出口局部高压的强度随着弹体初速度的增加而增大,随着水域长度的增加而不断减小。根据所受冲击载荷特性的不同,将前、后板分别划分为3个不同的区域,并建立了每个分区的简化计算模型。     

A new analytical model for the elastic–plastic behaviour of spot welded joints subjected to orthogonal load

An analytical procedure for the evaluation of the elastic–plastic stiffness behaviour of spot welded joints is presented. The procedure is based on a new model of spot weld region: a circular plate having variable thickness with a central rigid nugget, which is resolved using an original analytical method.The closed-form solution allows to describe the displacement of a rigid nugget when an axial orthogonal load is applied on the plate while plasticity and large deflections are present. The goal is to reach a reliable spot weld region model which can be used as the basis to develop a spot weld element in FE analysis even when plasticity and large deflections are in effect.The procedure is as completely original as no other can be found in the technical literature, and it has been applied to some examples of plates usually employed for spot weld analysis. The analytical results obtained by using the new general relations precisely match those obtained modelling spot weld area by FEA.     

An Application of the Kane and Mindlin Theory to Crack Problems in Plates of Arbitrary Thickness

Classical plane solutions of the theory of elasticity, which are sometimes more than 100 years old, are still used today and provide a framework for the analysis of many practical problems. But, strictly speaking, these analytical solutions are only applicable to plates with vanishing thickness or infinite thickness, where the stress state could be classified as plane stress or plane strain, respectively. However, the through-the-thickness stresses that exist in a plate of given thickness have a significant impact in a number of practical applications; and these stresses are often inevitably ignored due to the lack of analytical tools. This paper presents new analytical results for crack tip opening displacement (CTOD) for the through-the-thickness crack in infinite plates with various thicknesses. These results are based on the solution for an edge dislocation in infinite plate of arbitrary thickness and an application of the distributed dislocation technique. The analytical predictions of the CTOD and the constraint factor are compared with the three-dimensional elasto-plastic finite element (FE) results. It is shown that both analytical and numerical results are in good agreement when the numerical calculations are not affected by the size of the FE mesh and by the boundaries of the FE model.     

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.     

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.     

Shock and acceleration waves with large amplitudes in laminated composite plates

Propagation of shock and acceleration waves with large amplitudes is studied. The geometrical nonlinearity in the von Karman sense is included in deriving the plate equations. The dynamical conditions on the wave fronts are derived from the three-dimensional conditions in a way consistent with the derivation of the plate equations. General equations governing the propagation velocities are obtained. Solutions are presented for the case where the plates are initially at rest. It is found that, in this case, the large amplitude has a substantial effect only on the transverse shear shock wave. Finally, stability of the wave front is discussed.     

Modeling of failure wave propagation in impact compression of brittle materials (Glasses)

For a plane brittle failure wave in a material loaded by an elastic compression wave, an analytic solution is obtained on the basis of a model with a Drucker-Prager type condition and a given value of the failure wave propagation velocity. The wave failure model supplemented with threshold criteria is generalized to two-dimensional flows in breaking plates. Collision of glass plates with a rigid wall is modelled and three- and two-wave failure structures in the plate are obtained. Some data on the leading elastic wave decay and the arrest of the failure wave under the action of the overtaking unloading that propagates from the region of spread of the fractured material near the wall are obtained.     

Generalized closed-form model for analysis of asymmetric shaped charges

This paper presents a model that has less constraints than similar models and explains the collapse phenomenon in any desired order of geometrical asymmetries and in the presence of symmetric and asymmetric general-form wave fronts. It seems that, in this model, a complete generalized form of the classical jet formation theory has been developed. Available models that describe the symmetric jet and slug formation phenomenon are very good in such conditions. But the liner and confinement asymmetries, detonation wave front asymmetries, and other specifications, such as manufacturing tolerances, can affect the collapse and the behavior of the jet and slug. Some proposed models that describe asymmetric cases are not closed-form models or are only applicable for limited conditions, such as small asymmetries and a planar wave front. With the presented model, effects of concave, plane, and convex wave fronts on the off-axis velocity of the jet, other parameters of the jet and slug, and effects of an asymmetric wave front on jet formation for a completely symmetric liner and confinement geometry can be evaluated.     

The reflected pressure field in the interaction of weak shock waves with a compressible foam

This paper deals with the waves that are reflected from slabs of porous compressible foam attached to a rigid wall when impacted by a weak shock wave. The interest is in establishing possible attenuation of the pressure field after a shock or blast wave has struck the surface. Foam densities from 14 to 38 kg/m³ were tested over a range of shock wave Mach numbers less than 1.4. It is shown that the initial reflected shock wave strength is accurately predicted by the pseudo-gas model of Gelfand et al. (1983), with a pressure ratio of approximately 80% of the value for reflection off a rigid wall. Evidence is presented of gas entering the foam during the early stages of the process. A second wave emerges from the foam at a later stage and is separated from the first by a region of constant velocity and pressure. This second wave is not a shock wave but a compression front of significant thickness, which emerges from the foam earlier than predicted by the pseudo-gas model. Analysis of the origin of this wave points to much more complex flows within the foam than previously assumed, particularly in an apparent decrease in average wave front speed as the foam is compressed. It is shown that the pressure ratio across both these waves taken together is slightly higher than that for reflection off a rigid wall. In some cases this compression wave train is followed by a weak expansion wave.This article was processed using Springer-Verlag T_EX Shock Waves macro package 1990.     

缝水压力和裂缝面接触条件对重力坝裂缝扩展影响

基于多边形比例边界有限元模拟了重力坝裂缝扩展过程,综合考虑了坝-库水-地基相互作用、裂缝面接触和缝水压力的影响。其中该接触模型能够有效防止裂缝面的相互嵌入;该缝水模型除能考虑裂缝面开合速度、缝隙宽度和水流形态对缝水压力分布的影响,还可基于连续性条件模拟缝内水体流动和空化的现象。以Koyna坝裂缝扩展为例,研究了裂缝面接触条件和缝水压力对计算结果的影响。结果显示,考虑接触的影响,震损较为严重。而与假定为常水压力分布的情况相比,实际的缝水压力分布形式对结果影响较为复杂。在裂缝张开时,减小了裂缝扩展的可能,而在裂缝闭合时正好与之相反。