Analytical Solution for Isothermal Flow in a Shock Tube Containing Rigid Granular Material

Analytical solution of shock wave propagation in pure gas in a shock tube is usually addressed in gas dynamics. However, such a solution for granular media is complex due to the inclusion of parameters relating to particles configuration within the medium, which affect the balance equations. In this article, an analytical solution for isothermal shock wave propagation in an isotropic homogenous rigid granular material is presented, and a closed-form solution is obtained for the case of weak shock waves. Fluid mass and momentum equations are first written in wave and (mathematical) non-conservation forms. Afterwards by redefining the sound speed of the gas flowing inside the pores, an analytical solution is obtained using the classical method of characteristics, followed by Taylor’s series expansion based on the assumption of weak flow which finally led to explicit functions for velocity, density and pressure. The solution enables plotting gas velocity, density and pressure variations in the porous medium, which is of high interest in the design of granular shock isolators.     

Mechanism of compressive stress formation during weak shock waves impact with granular materials

 Certain aspects of wave propagation and the dynamic reaction of a granular material when subjected to a long-duration impulse load are studied. In the majority of studies published on this subject the unsteady pressure behavior at the end-wall covered by a layer of granular material was observed and documented. However, up to now little attention was given to explaining the physical mechanism of this process. Experimental results, obtained in the course of this study, regarding the pressure fields inside granular layers of different materials, clearly show that the compaction effect strongly depends on the characteristics of the medium. This phenomenon manifests itself by changing the gas-particle interaction in the course of the gas filtration, and by variation in the contribution of the different forces and effective stress, σ, to the energy exchange between the gas, the particles and the shock-tube wall. The material permeability,  f, the relative density, ν, and the particle response time, τ _p , are the most important parameters affecting the stress formation at the end-wall covered by the granular layer. In addition to the effect of the material parameters, the effective stress, σ, was found to strongly depend on the granular layer height, h. Based on detailed pressure measurements a qualitative analysis regarding the role of the particle rearrangement in the formation of the unsteady peak at the end-wall was performed. The phenomenology of the particle–particle interaction includes rotation and consolidation of the granules and movement or sliding of particle planes within the layer over each other. Most of these processes are frictional in their nature. They are related to the energy losses and affect the profile and magnitude of the compressive stress as measured at the shock-tube end-wall covered by the granular layer. Received: 10 June 1996/Accepted: 15 October 1996     

Experimental investigation of the interaction between weak shock waves and granular layers

 The paper describes new experimental results regarding the pressure fields in front of and inside granular layers of different materials during their collision with weak shock waves. A variety of waves result from the shock wave-granular layer interaction. The pressure behind the reflected wave from the material interface approaches the equilibrium value, P ₅, which would have been reached had the shock wave reflected from a solid end-wall. The wave succession inside the layer depends solely on two processes: the complex interaction of the compaction wave with the granular material and the gas filtration, which affects the particles by the drag forces between the two phases. Inside a material with a permeability coefficient f>0.001 mm² the transmitted wave moves with a constant velocity which is largely governed by the gas filtration. For low permeability materials ( f<0.0003 mm²) the transmitted wave trajectory strongly depends on the compaction wave propagation. In such cases the compaction wave was found to be unsteady and its acceleration was higher in material having low material densities. The maximum compressive stress values, P _c , reached at the shock tube end-wall, covered by the materials under investigation, manifested as an unsteady pressure peak twice as large as the gas pressure P ₅, measured ahead of the layer. Comparing the present data with those available in the literature showed that the amplitude of the unsteady pressure peak was higher in materials having low effective densities, γ, and small permeability coefficients f. Contrary to flexible foams where the available experimental data indicated that the compressive stress in the post peak period converges to P ₅=P _g , the results obtained in the present study indicated that during the test time the compressive stress, P _s , was well preserved in the material and for most of the sample length its value was within the range P _s >P ₅>P _g . Received: 4 March 1996 / Accepted:26 September 1996     

Behavior of shock trains in a diverging duct

A shock train inside a diverging duct is analyzed at different pressure levels and Mach numbers. Nonreactive pressurized cold gas is used as fluid. The structure and pressure recovery inside the shock train is analyzed by means of wall pressure measurements, Schlieren images and total pressure probes. During the course of the experiments, the total pressure of the flow, the back pressure level and the Mach number upstream of the compression region have been varied. It is shown that the Reynolds number has some small effect on the shock position and length of the shock train. However, more dominant is the effect of the confinement level and Mach number. The results are compared with analytical and empirical models from the literature. It was found that the empirical pseudo-shock model from Billig and the analytical mass averaging model from Matsuo are suitable to compute the pressure gradient along the shock train and total pressure loss, respectively.     

Parametric analysis of the regimes of shock-wave action on gas-liquid media

The salient features of shock and isentropic action on gas-liquid media are investigated using a wide-range equation of state for water and vapor. The effect of the pressure and the vapor (gas) content on the speed of sound in the gas-liquid mixture is considered. The parameters of incident and reflected wave in the gas-liquid medium are obtained on the basis of the Rankine-Hugoniot relations for the cases of isothermal, adiabatic, and shock compression of the gas component. It is shown that when the Rankine-Hugoniot equations of the shock compression of the mixture are used within the framework of the single-velocity, two-temperature model with the same pressure and under the condition of the additivity-in-mass of the internal energy of the mixture, each fraction is compressed in accordance with its own individual shock adiabat equation. The calculated and experimental data on the acoustic and shock wave propagation in vapor- and gas-liquid media and their reflection from barriers are compared.     

Numerical simulations of shock-induced load transfer processes in granular media using the discrete element method

By the discrete element method (DEM), we perform numerical simulations of shock-induced load transfer processes in granular layers composed of spherical particles packed in vertical channels. In order to isolate the load transfer through the grains’ contact points from the complicated load transfer processes, we simulate the shock wave interactions with granular layers having no permeability for gas. The shock loading is achieved by applying a downward step force on the top of the granular layers. Complex, three-dimensional load transfer processes in the granular media, which are extremely difficult to understand from experiments, are visualized based on the results from the present DEM simulation. The numerical results show that highly concentrated load transfer paths, through which shock loads are transferred mainly, exist in the granular media, and that the dimensions of the container of the granular media considerably affect the shock-induced load transfer processes. From a coarse-grained representation of intergranular stress, wave-like load transfer processes are clearly observed. For relatively deep granular layers, however, the wave fronts became unclear as they propagated.     

水下爆破冲击波作用下典型鱼类临界安全波压模型研究

为了探究水下爆破冲击波对鱼类损伤的影响,首先通过理论分析探讨了水下冲击波在鱼体的传播过程和对鱼鳔的损伤机理,构建了典型有鳔鱼类临界安全波压模型,并结合实测数据、参考相关文献,确定了模型参数。结果表明,典型有鳔鱼类临界安全波压与鱼体长呈线性关系。其中,水介质和鱼鳔壁介质的波阻抗比、鱼鳔宽度因数、鱼鳔壁厚因数、鱼鳔径向临界拉应力因数和鱼鳔形状因数分别为0.3～2.0、0.04～0.09、0.002、60和0.6～1.1。然后,根据典型有鳔鱼类临界安全波压模型参数,得到最大和最小鱼类临界安全波压模型分别为p_ic,max=30L和p_ic,min=3L(p_ic,max的单位为kPa,L的单位为cm)。最后,通过鱼类损伤情况的实测数据和文献数据,验证了鱼类临界安全波压模型。结果表明,不同体长的鱼类在不同冲击波压力时的受损状况分布,基本与鱼类所能承受的最大和最小的临界安全波压范围符合。并根据鱼类临界安全波压与体长的关系,划分了死亡区、存活有影响区和存活无影响区。     

酚醛层压材料的冲击力学行为及本构模型

为了研究酚醛层压材料的冲击力学行为并获得本构模型,利用万能试验机和整形修正的分离式霍普金森压杆(SHPB)装置,对材料试样进行了应变率范围为10-3~103 s-1的单轴压缩实验,得到了不同加载应变率下的应力应变曲线,对其在准静态、动态载荷下的压缩破坏机理进行了初步探讨。结果表明,酚醛层压材料具有较强的应变率效应,与准静态(1.67×10-3 s-1)时相比,在动态载荷(7×102 s-1)下,峰值应力增加了约10倍;破坏应变减少了约一半;在准静态和动态加载条件下试样力学性能的差异是由于纤维基体界面特性以及不同应变率下破坏模式的不同;采用朱-王-唐本构方程描述了酚醛层压材料力学行为,拟合得到了本构方程的系数,在加载过程中,理论计算值与实验结果吻合较好。     

Incorporating the effects of fabric in the dilatant double shearing model for planar deformation of granular materials

The objective of this paper is to incorporate the effects of fabric and its evolution into the Dilatant Double Shearing Model [Mehrabadi, M.M., Cowin, S.C., 1978. Initial planar deformation of dilatant granular materials. J. Mech. Phys. Solids 26, 269–284] for granular materials in order to capture the anisotropic behavior and the complex response of granular materials in cyclic shear loading. An important consequence of considering the fabric is that one can have unequal shearing rates along the two slip directions. This property leads to the non-coaxiality of the principal axes of stress and strain rate, which is more appropriate for a material that exhibits initial and induced anisotropy. In addition, we employ a fabric-dependent elasticity tensor with orthotropic symmetry. The model developed in this paper also predicts one of the experimentally observed characteristics of granular materials: the gradual concentration of the contact normals towards the maximum principal stress direction.We implement the constitutive equations into ABAQUS/Explicit by writing a user material subroutine in order to predict the strength anisotropy of granular materials in a plane strain biaxial compression test and investigate the mechanical behavior of granular materials under the cyclic shear loading conditions. The predictions from this model show good quantitative agreement with the experiments of [Park, C.S., 1990. Anisotropy in deformation and strength properties of sands in plane strain compression, Masters Thesis, University of Tokyo; Park, C.S., Tatsuoka, F., 1994. Anisotropic strength and deformation of sands in plane strain compression. In: XIII ICSMFE, New Delhi, India; Okada, N., 1992. Energy dissipation in inelastic flow of cohesionless granular media. Ph.D. Thesis, University of California, San Diego].     

含电子相影响的多功能含能结构材料冲击压缩理论计算

为了更好地描述疏松态金属材料的冲击压缩特性,基于托马斯-费米原子统计模型,研究金属晶体中电子热行为对系统内粒子数、内能、压强等参数的影响,修改了描述疏松金属材料的Wu-Jing模型中的参数R的计算方法。结合混合物的冷能叠加原理,得到考虑电子相影响的疏松态混合物物态方程。并对不同配比的密实态W/Cu合金、不同疏松度的Al/Ni合金的典型多功能含能结构材料进行计算,获得其冲击压力-比容关系及冲击波速度-粒子速度关系,计算结果与实验结果吻合较好。结果表明,本文中模型对未反应条件下的金属材料冲击压缩特性预测较好;疏松材料的冲击压力-粒子速度关系并不呈现出密实材料的近似线性关系,其冲击压缩过程分为压实前和压实后2个明显的阶段;多功能含能结构材料的冲击压缩特性受材料孔隙率、材料配比等影响明显。     

Thermoelastic Couplings and Interparticle Friction Evidenced by Infrared Thermography in Granular Materials

Infrared (IR) thermography was used to analyze the thermomechanical response of a two-dimensional non-cohesive granular assembly. Two constitutive materials with different types of thermoelasticity were chosen: thermoplastic polyurethane (TPU) and polyoxymethylene (POM), which feature entropic and isentropic elasticity respectively. Cylinders of each material were mixed together. Analysis was performed under confined compression at two observation scales. Thermoelastic couplings and interparticle friction were separately evidenced. First, the strong thermal effect of entropic coupling was revealed at the contacts, in the stress concentration zones. Second, image processing enabled us to clearly extract the thermal signature of the interparticle friction zone, a quantity that cannot be identified by the other full-field measurement techniques available today. It can thus be claimed that IR thermography provides two distinct routes for the analysis of granular materials by distinguishing the reversible and irreversible parts from the global thermomechanical response. The study also opens prospects for the experimental analysis of “soft” granular media.     

基于分子动力学模拟的单晶硅冲击压缩相变研究

晶体硅具有复杂的相变机制,在相图研究中受到广泛关注,其在动载荷下的变形机制是当前研究热点。为揭示晶体硅在强动加载下的变形和相变行为特征,基于分子动力学方法,采用平板冲击加载方式,模拟研究了单晶硅在初始环境温度为300 K时分别沿[001]、[110]和[111]晶向的不同强度下的冲击压缩行为,冲击粒子速度为0.3～3.2 km/s。研究发现,随着冲击粒子速度的增加,单晶硅剪切应力在逐渐增加后由于结构相变发生急剧下降,相变阈值和相变机制均呈现各向异性。其中,沿[001]晶向冲击压缩下观察到多种固-固相变以及固-液相变,并观察到与最新文献的实验高度一致的固-液共存现象。研究结果可为动加载下晶体硅的相变研究提供纳米尺度的结果支撑。     

Evolution of blast wave profiles in simulated air blasts: experiment and computational modeling

Shock tubes have been extensively used in the study of blast traumatic brain injury due to increased incidence of blast-induced neurotrauma in Iraq and Afghanistan conflicts. One of the important aspects in these studies is how to best replicate the field conditions in the laboratory which relies on reproducing blast wave profiles. Evolution of the blast wave profiles along the length of the compression-driven air shock tube is studied using experiments and numerical simulations with emphasis on the shape and magnitude of pressure time profiles. In order to measure dynamic pressures of the blast, a series of sensors are mounted on a cylindrical specimen normal to the flow direction. Our results indicate that the blast wave loading is significantly different for locations inside and outside of the shock tube. Pressure profiles inside the shock tube follow the Friedlander waveform fairly well. Upon approaching exit of the shock tube, an expansion wave released from the shock tube edges significantly degrades the pressure profiles. For tests outside the shock tube, peak pressure and total impulse reduce drastically as we move away from the exit and majority of loading is in the form of subsonic jet wind. In addition, the planarity of the blast wave degrades as blast wave evolves three dimensionally. Numerical results visually and quantitatively confirm the presence of vortices, jet wind and three-dimensional expansion of the planar blast wave near the exit. Pressure profiles at 90° orientation show flow separation. When cylinder is placed inside, this flow separation is not sustained, but when placed outside the shock tube this flow separation is sustained which causes tensile loading on the sides of the cylinder. Friedlander waves formed due to field explosives in the intermediate-to far-field ranges are replicated in a narrow test region located deep inside the shock tube.     

反射激波作用重气柱的Richtmyer-Meshkov不稳定性的实验研究

采用高速摄影结合激光片光源技术,研究了反射激波冲击空气环境中重气体(SF₆)气柱的Richtmyer-Meshkov不稳定性。通过在横式激波管试验段采用可移动反射端壁获得不同反射距离,实现了反射激波在不同时刻二次冲击处于演化中后期的气柱界面,得到了不同的界面演化规律。反射距离较小时,斜压机制对气柱界面形态演化的影响显著,界面衍生出二次涡对结构;反射距离较大时,压力扰动机制的影响显著,界面在流向上被明显地压缩,没有形成明显的涡结构。由气柱界面形态的时间演化图像得到了界面位置和整体尺度随时间的变化,对反射激波作用后气柱界面的演化进行了量化分析。     

Dense particle cloud dispersion by a shock wave

A dense particle flow is generated by the interaction of a shock wave with an initially stationary packed granular bed. High-speed particle dispersion research is motivated by the energy release enhancement of explosives containing solid particles. The initial packed granular bed is produced by compressing loose powder into a wafer with a particle volume fraction of $\phi _\mathrm{p} = 0.48$ . The wafer is positioned inside the shock tube, uniformly filling the entire cross-section. This results in a clean experiment where no flow obstructing support structures are present. Through high-speed shadowgraph imaging and pressure measurements along the length of the channel, detailed information about the particle shock interaction was obtained. Due to the limited strength of the incident shock wave, no transmitted shock wave is produced. The initial solid-like response of the particle wafer acceleration forms a series of compression waves that eventually coalesce to form a shock wave. Breakup is initiated along the periphery of the wafer as the result of shear that forms due to the fixed boundary condition. Particle breakup is initiated by local failure sites that result in the formation of particle jets that extend ahead of the accelerating, largely intact, wafer core. In a circular tube, the failure sites are uniformly distributed along the wafer circumference. In a square channel, the failure sites, and the subsequent particle jets, initially form at the corners due to the enhanced shear. The wafer breakup subsequently spreads to the edges forming a highly non-uniform particle cloud.     

Effect of particle size distribution on micro- and macromechanical response of granular packings under compression

The role of particle size heterogeneity on micro- and macromechanical properties of assemblies of spherical particles was studied using DEM simulations. The response to an imposed load of a granular material composed of non-uniformly sized spheres subjected to uniaxial confined compression was investigated. A range of geometrical and micro-mechanical properties of granular packings (e.g., void fraction, contact force distribution, average coordination number and degree of mobilisation of friction at contacts between particles) were examined, and provided a more accurate interpretation of the macroscopic behaviour of mixtures than has previously been available. The macromechanical study included stress transmission, stiffness and angle of internal friction of the granular assemblies.The degree of polydispersity showed slight effect on both, the void fraction and the elastic properties of the system. The tendency for increase in the lateral-to-vertical pressure ratios was observed with an increasing degree of particle size heterogeneity; however, the different pressure ratios calculated for samples with various degrees of polydispersity lay within the range of data scatter.     

Numerical simulation study on particle breakage behavior of granular materials in confined compression tests

In order to study the fragmentation law, the confined compression experiment of granular assemblies has been conducted to explore the particle breakage characteristic by DEM approach in this work. It is shown that contact and contact force during the loading process gradually transform from anisotropy to isotropy. Meanwhile, two particle failure modes caused by different contact force states are analyzed, which are single-through-crack failure and multi-short-crack failure. Considering the vertical distribution of the number of cracks and the four characteristic stress distributions (the stress related to the maximum contact force, the major principal stress, the deviatoric stress and the mean stress), it is pointed out that the stress based on the maximum contact force and the major principal stress can reflect the distribution of cracks accurately. In addition, the size effect of particle crushing indicates that small size particles are prone to break. The lateral pressure coefficient of four size particles during the loading process is analyzed to explain the reason for the size effect of particle breakage.     

Gas pressure equalization in a porous medium filling a pipe with a closed end under shock loading

The problem of gas pressure equalization in a porous medium filling a pipe with a closed end under shock loading is solved. In this case, the initial filtration velocity behind the shock wave should be specified as initial data, in addition to the shock pressure. It is shown that the shock decays at a finite distance and pressure equalization occurs in a finite time. Approximate submodels of discontinuous and smooth solutions are obtained.     

含瓦斯煤单轴压缩的尺度效应实验研究

含瓦斯煤与其它岩石类材料一样,在力学特性等方面具有尺度效应。利用特制的饱和加压装置和电子万能试验机,对不同尺度、不同饱和瓦斯压力下的煤样进行了单轴压缩试验,探讨了煤样变形破坏及强度特性的变化规律。分析表明,煤样抗压强度、弹性模量随瓦斯压力的增大而减小,抗压强度随高径比的增加明显减小,弹性模量随高径比的增加略有提高。利用Weibull模型和线弹性强化理论对含瓦斯煤产生尺度效应的原因进行了合理地解释,为含瓦斯煤样尺度效应在强度表征等方面的深入研究提供了新的实验和理论依据。     

气压对压力容器前壁超高速撞击损伤的影响

充气压力容器在超高速撞击下的典型损伤包括穿孔及其边缘的裂纹失稳破坏,会导致气体泄漏或爆炸,内压对容器前壁损伤的影响仍不明确。以不同内压的球形铝合金充气压力容器为研究对象,开展了球形铝合金弹丸超高速撞击实验和数值模拟计算,分析了内充气体压强对前壁穿孔形貌特征、穿孔直径、孔边环向应力等的影响规律和影响机理,讨论了气体冲击波的传播行为及影响前壁穿孔边缘裂纹失稳破坏的机制。结果表明:前壁穿孔边缘内翻边形貌与内压相关,内压越高,弯折程度越轻;穿孔直径与内充气体压强正相关,但气体对孔径的影响远小于容器壁厚及撞击速度的影响;穿孔边缘使裂纹失稳破坏的环向拉应力不仅受到后壁反射冲击波的影响,也与容器壁内应力波的传播有关,与内压成正比。