Plane wave propagation in 2D and 3D monodisperse periodic granular media

Plane wave propagation in periodic ordered granular media comprising of elastic spherical particles is investigated. The spheres are under zero precompression and are assumed to interact via the Hertzian contact potential. Various two- and three-dimensional granular structures such as hexagonal packing (2D and 3D), face-centered cubic and body-centered cubic packings are considered in the present study, with the plane impact either normal or oblique to the granular system. For the normal impact case, 1D chains equivalent to the 2D and 3D structures are obtained. A universal relation between the wavefront speed and the force amplitude is derived, valid for all the granular structures studied. In the angular impact case, the shear component of the amplitude of the particle velocity is found to initially decay exponentially and further in a series of linear regimes. By employing simpler models, semi-analytical predictions are obtained for the decay of shearing effect.     

Attenuation or enhancement—a one-dimensional analysis on shock transmission in the solid phase of a cellular material

The characteristics of compressive shock wave propagation in the solid phase of a cellular material are studied in the present paper using a one-dimensional mass-spring model. The unique compressive stress–strain relation of a cellular material leads to several interesting observations on the characteristic of one-dimensional stress wave transmission in a cellular material, which are important for understanding the blast and impact mitigation and attenuation through a cellular material. Generally, cellular material attenuates impact- or blast-induced loads by cell collapse mechanism at low impact velocities or low pulse pressure intensities when the stress transmission in a cellular material is limited by the plateau stress before the densification stage starts. This feature leads to wide applications of cellular materials in structural crashworthiness design where low speed impact is considered as potential survivable scenarios. However, scattered information has shown that stress enhancement in cellular material may occur when an intensive loading is applied, which, in contrast to the stress attenuation function of a cellular material, could produce more severe damage on the protected structures. This phenomenon is studied qualitatively in the present paper using a one-dimensional spring-mass model.     

Numerical simulation of the guided Lamb wave propagation in particle reinforced composites

This paper deals with the investigation of the Lamb wave propagation in particle reinforced composites excited by piezoelectric patch actuators. A three-dimensional finite element method (FEM) modeling approach is set up to perform parameter studies in order to better understand how the Lamb wave propagation in particle reinforced composite plates is affected by change of central frequency of excitation signal, volume fraction of particles, size of particles and stiffness to density ratio of particles. Furthermore, the influence of different arrangements is investigated. Finally, the results of simplified models using material data obtained from numerical homogenization are compared to the results of models with heterogeneous build-up. The results show that the Lamb wave propagation properties are mainly affected by the volume fraction and ratio of stiffness to density of particles, whereas the particle size does not affect the Lamb wave propagation in the considered range. As the contribution of the stiffer material increases, the group velocity and the wave length also increase while the energy transmission reduces. Simplified models based on homogenization technique enabled a tremendous drop in computational costs and show reasonable agreement in terms of group velocity and wave length.     

Dynamic frictional slip along an interface between plastically compressible solids

Dynamic frictional slip along an interface between plastically compressible solids is analyzed. The plane strain, small deformation initial/boundary value problem formulation and the numerical method are identical to those in Shi et al. (Int J Fract 162:51, 2010) except that here the material constitutive relation allows for plastic compressibility. The interface is characterized by a rate and state dependent friction law. The specimens have an initial compressive stress and are subject to shear loading by edge impact near the interface. Two loading conditions are analyzed, one giving rise to a crack-like mode of slip propagation and the other to a pulse-like mode of slip propagation. In both cases, the initial compressive stress is taken to vary with plastic compressibility such that the associated initial effective stress is the same for all values of plastic compressibility. The volume change for the crack-like slip mode is mainly plastic while the elastic volume change plays a larger role for the pulse-like mode. For the crack-like slip mode, the proportion of plastic dissipation in the material increases with the increasing plastic compressibility, but the effect of plastic compressibility on the energy partitioning for the pulse-like slip mode is much smaller. The predicted propagation speeds approach a speed about the dilational wave speed for both the crack-like and pulse-like slip modes and this speed is not sensitive to the value of the plastic compressibility parameter. Plastic dissipation is found to be mainly associated with the deformation induced by the loading wave rather than with the deformation arising from slip propagation. The amplitude of the slip rate in the slip pulses is found to be largely governed by the value of the initial compressive stress regardless of the value of plastic compressibility.

     

Sliding along frictionally held incoherent interfaces in homogeneous systems subjected to dynamic shear loading: a photoelastic study

An experimental investigation was conducted to study dynamic sliding at high strain rates along incoherent (frictional) interfaces between two identical plates. The plates were held together by a uniform compressive stress, while dynamic sliding was initiated by an impact-induced shear loading. The case of freely-standing plates with no external pressure was also investigated. The dynamic stress fields that developed during the events were recorded in a microsecond time scale by high-speed photography in conjunction with classical dynamic photoelasticity. Depending on the choice of experimental parameters (impact speed and superimposed static pressure), pulse-like and crack-like sliding modes were observed. Visual evidence of sub-Rayleigh, intersonic and even supersonically propagating pulses were discovered and recorded. Unlike classical shear cracks in coherent interfaces of finite strength, sliding areas in frictional interfaces seem to grow at various discrete speeds without noticeable acceleration phases. A relatively broad loading wave caused by the interference between the impact wave and the preexisting static stress field was observed emanating from the interface. There was a cusp in the stress contours at the interface, indicating that the propagation speed was slightly faster along the interface than in the bulk. The observed propagation speeds of the sliding tips were dependent on the projectile speed. They spanned almost the whole interval from sub-Rayleigh speeds to nearly the sonic speed of the material, with the exception of a forbidden gap between the Rayleigh wave speed and the shear wave speed. Supersonic trailing pulses generating Mach lines of different inclination angles, emanating from the sliding zone tips, were discovered. In addition, behind the sliding tip, wrinkle-like opening pulses were observed for a wide range of impact speeds and confining stresses. They always traveled at speeds between the Rayleigh wave speed and the shear wave speed of the material. Symposium on Physics and Scaling in Fracture held during the ICF11 (2005) in Turin.     

Physical processes within a 2D granular layer during an impact

In this paper, the impact of a block on a coarse granular soil corresponding to rockfall events is investigated using the Discrete Element Method. Different impacting particle and medium characteristics (impact point, impacting particle size and shape, sample height, etc.) are considered. The numerical results first exhibit the physical phenomena involved in the interaction between the impacting particle and the granular medium. The impact process starts with the partial energy exchange from the impacting particle to the soil. This phase is followed by the propagation of a shockwave from the impact point and a wave reflection on the bottom wall of the sample. A second energy exchange from soil particles to the impacting particle can occur if the reflected wave reaches the soil surface before the end of the impact. Based on these investigations, the impacting particle bouncing occurrence diagram is defined for various impacting particle sizes, incident kinematic parameters and sample heights. The bouncing occurrence diagram brings out three impact regimes. For a small impacting particle, the impact is mainly determined by the first interaction between the impacting particle and the soil, whereas for an intermediate-sized impacting particle, the shockwave propagation through the sample is the leading phenomenon. For a large impacting particle, bouncing is associated with the formation of a compact layer below the impacting particle.     

Force propagation speed in a bed of particles due to an incident particle impact

The force propagation speed in granular matter is a very difficult property to be measured. A new technique has been developed to calculate the force propagation speed in granular matter based on measuring experimentally the contact time. The contact time for a particle hitting a bed of particles is estimated as the time taken for a particle to strike a bed of particles till the time of its ejection, and it is calculated using the discrete element method. The speed of force propagation in a bed of particles is estimated by plotting the dependence of the path length of the contact force on the contact time and finding the gradient of such dependence. Such approach leads to accurate results if the impact speed is below the yield velocity, i.e. no plastic deformations. It is found that the force propagation speed in spherical granular matter is proportional to the impact speed of the incident particle, which is different from force propagation in continuum matter. It is also found that the propagation speed is dependent on the material and diameters ratio of the interacting particles, but it is not dependent on the number of bed layers. The propagation speed in granular matter is normalized by dividing it by a reference propagation speed, i.e. the propagation speed at an impact speed of 1 m/s. It is found that the normalized propagation speed is independent of the material and diameter of the interacting particles, but it is logarithmically proportional to the impact speed. The proportionality constant is equal to 0.16, which can be taken as a universal constant for force propagation in spherical granular matter.     

Analysis of the dispersion relation for an incompressible transversely isotropic elastic plate

Summary The dispersion relation associated with harmonic wave propagation in an incompressible, transversely isotropic elastic plate is derived. Such a material is characterized by only three material constants, contrasting with five in the corresponding compressible case. Motivated by a numerical investigation, asymptotic expansions, giving phase speed and frequency as functions of wave number, are derived in both the long and short wave regimes. These approximations, which owing to the constitutive simplifications are readily available, are shown to provide excellent agreement with the corresponding numerical solution. It is envisaged that the detailed investigation carried out in this paper will aid numerical inversion of the transform solutions often used in impact problems. Additionally, the asymptotic investigation provides the necessary basis for future studies to derive asymptotically approximate models to describe long and short wave motion.     

Failure of cemented granular materials under simple compression: experiments and numerical simulations

We investigate the strength and failure properties of a model cemented granular material under simple compressive deformation. The particles are lightweight expanded clay aggregate beads coated by a controlled volume fraction of silicone. The beads are mixed with a joint seal paste (the matrix) and molded to obtain dense cemented granular samples of cylindrical shape. Several samples are prepared for different volume fractions of the matrix, controlling the porosity, and silicone coating upon which depends the effective particle–matrix adhesion. Interestingly, the compressive strength is found to be an affine function of the product of the matrix volume fraction and effective particle–matrix adhesion. On the other hand, it is shown that particle damage occurs beyond a critical value of the contact debonding energy. The experiments suggest three regimes of crack propagation corresponding to no particle damage, particle abrasion and particle fragmentation, respectively, depending on the matrix volume fraction and effective particle–matrix adhesion. We also use a sub-particle lattice discretization method to simulate cemented granular materials in two dimensions. The numerical results for crack regimes and the compressive strength are in excellent agreement with the experiments.     

Shock dynamics in granular chains: numerical simulations and comparison with experimental tests

The aim of this paper is to simulate the nonlinear wave propagation in granular chains of beads using a recently introduced multiple impact model and to compare numerical results to experimental ones. Different kinds of granular chains are investigated: monodisperse chains, tapered chains and stepped chains. Particular attention is paid to the dispersion effect, and the wave propagation in tapered chains, the interaction of two solitary waves in monodisperse chains, and the formation of solitary wave trains in stepped chains. We show that the main features of the wave propagation observed experimentally in these granular chains are very well reproduced. This proves that the considered multiple impact model and numerical scheme are able to encapsulate the main physical effects that occur in such multibody systems.     

Amplitude-dependent attenuation of compressive waves in curved granular crystals constrained by elastic guides

We study the wave propagation in a curved chain of spherical particles constrained by elastic guides under the axial impact of a falling mass. We characterize the force transmission properties of the chain by varying the striker’s mass and the chain’s curvature. Experimental tests demonstrate amplitude-dependent attenuation of compressive waves propagating through the curved chain. In particular, we observe that the curved systems present an improved transmission of small dynamic disturbances relative to that of strong excitations, resulting from the close interplay between the granular particles and the softer elastic medium. We also find that the transmission of the compressive waves through the chains is dependent on the initial curvature imposed to the system. Numerical simulations, based on an approach that combines discrete element and finite element methods, corroborate the experimental results. The findings suggest that hybrid structures composed of granular particles and linear elastic media can be employed as new passive acoustic filtering materials that selectively transmit or mitigate excitations in a desired range of pressure amplitudes.     

爆炸载荷下空孔缺陷与爆生裂纹扩展行为研究

空孔在岩石巷道直眼掏槽爆破中具有重要作用,为研究空孔及其缺陷在爆炸荷载作用下的扩展行为和作用机理,以PMMA代替岩石材料,利用预制裂纹代替空孔缺陷,借助动态焦散线系统和理论分析为手段,研究不同间距下空孔、空孔处预制裂纹、爆生裂纹动态扩展规律及机理,分析不同"径距比"与掏槽效果的关系。研究结果表明:在装药量一定的情况下,随着炮孔与空孔距离的变化,爆生裂纹扩展距离呈现递增而预制裂纹扩展距离呈现递减的趋势,但都存在极值;当炮孔与空孔距离较小时,爆生裂纹和预制裂纹扩展及相互作用最复杂,爆生裂纹扩展经历由压缩应力波为主,表现为直线的前期扩展;由空孔处发射应力波和压缩应力波共同作用下,爆生裂纹偏离炮孔与空孔连心线的中期扩展,以及由空孔应力集中区作用使爆生裂纹向空孔方向偏移的后期阶段;预制裂纹扩展经历由空孔处应力集中作用下,表现为直线的前期扩展,以及由爆生裂纹处反射拉伸波作用使其向爆生裂纹发展的后期阶段;当炮孔与空孔距离较大时,反射应力波及应力集中效应对爆生裂纹和预制裂纹扩展在减弱,爆生裂纹与预制裂纹扩展行为仅有前期直线扩展阶段。"径距比"的大小对爆破效果影响较大,直眼掏槽爆破应以最优"径距比"作为掏槽爆破参数设计的依据。     

Explicit dynamics in impact simulation using a NURBS contact interface

In this paper, the impact problem and the subsequent wave propagation are considered. For the contact discretization an intermediate non-uniform rational B-spline (NURBS) layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient element-based integration. The impact event is ill-posed and requires a regularization to avoid propagating stress oscillations. A nonlinear mesh-dependent penalty regularization is used, where the stiffness of the penalty regularization increases upon mesh refinement. Explicit time integration methods are well suited for wave propagation problems, but are efficient only for diagonal mass matrices. Using a spectral element discretization in combination with a NURBS contact layer the bulk part of the mass matrix is diagonal.     

On the effects of reflected waves in transient shear wave elastography

In recent years, novel quantitative techniques have been developed to provide noninvasive and quantitative stiffness images based on shear wave propagation. Using radiation force and ultrafast ultrasound imaging, the supersonic shear imaging technique allows one to remotely generate and follow a transient plane shear wave propagating in vivo in real time. The tissue shear modulus, i.e., its stiffness, can then be estimated from the shear wave local velocity. However, because the local shear wave velocity is estimated using a time-of- flight approach, reflected shear waves can cause artifacts in the estimated shear velocity because the incident and reflected waves propagate in opposite directions. Such effects have been reported in the literature as a potential drawback of elastography techniques based on shear wave speed, particularly in the case of high stiffness contrasts, such as in atherosclerotic plaque or stiff lesions. In this letter, we present our implementation of a simple directional filter, previously used for magnetic resonance elastography, which separates the forward- and backward-propagating waves to solve this problem. Such a directional filter could be applied to many elastography techniques based on the local estimation of shear wave speed propagation, such as acoustic radiation force imaging (ARFI), shearwave dispersion ultrasound vibrometry (SDUV), needle-based elastography, harmonic motion imaging, or crawling waves when the local propagation direction is known and high-resolution spatial and temporal data are acquired.     

基于应变空间的碾压混凝土各向异性损伤本构模型

根据碾压混凝土材料的力学特性和损伤拉压显著不同的特点,分别在拉应变和压应变空间建立了碾压混凝土的本构关系和损伤演化方程。在拉应变空间,碾压混凝土的变形特性表现为脆弹性,考虑弹性与损伤耦合,应用正交各向异性损伤理论描述碾压混凝土的刚度退化和应变软化;在压应变空间,考虑弹塑性与损伤耦合,应用内时理论来描述碾压混凝土的弹塑性特性,正交各向异性损伤理论来描述微裂缝扩展引起的刚度退化和应变软化,内时理论没有屈服面,使模型的参数和方程大大减少,从而简化了非线性计算过程。计算结果表明,该模型能够较好地描述碾压混凝土在单轴和多轴加载下的性质。     

Stability of dynamically propagating cracks in brittle materials

Summary Dynamic crack propagation and bifurcation phenomena are investigated analytically by utilizing the strain energy density fracture criterion in the framework of catastrophe theory. The effect of biaxial stress, loading imperfections (mixed-mode loading), Poisson's ratio, state of stress as well as crack tip propagation speed on the crack path directional stability is analyzed. Special crack path stability charts for (un)stably propagating cracks are obtained, and their connection with the experimentally recorded crack tip stress field is addressed. It is shown that a slight change of the normal stress acting parallel to a crack at its tip (crack-parallel stress) may be able to affect the crack surface roughening and/or branching velocity considerably. It is also indicated that under small tensile crack-parallel stress, the crack propagation is stable only when the crack propagation speed is less than about 30% of the relevant shear wave speed. The crack becomes unstable, and its surfaces roughen severely at a higher speed, and the crack bifurcates at the highest propagation speed, some 45% of the shear wave speed. It is suggested that superimposing mode-II (shear) loading will enhance the dynamic crack path stability while increasing crack propagation speed will reduce the stability of crack propagation. It is expected that under compressive crack-parallel stress no crack surface roughening will occur before the crack stably bifurcates.     

Influence of packing density and stress on the dynamic response of granular materials

Laboratory geophysics tests including bender elements and acoustic emission measure the speed of propagation of stress or sound waves in granular materials to derive elastic stiffness parameters. This contribution builds on earlier studies to assess whether the received signal characteristics can provide additional information about either the material’s behaviour or the nature of the material itself. Specifically it considers the maximum frequency that the material can transmit; it also assesses whether there is a simple link between the spectrum of the received signal and the natural frequencies of the sample. Discrete element method (DEM) simulations of planar compression wave propagation were performed to generate the data for the study. Restricting consideration to uniform (monodisperse) spheres, the material fabric was varied by considering face-centred cubic lattice packings as well as random configurations with different packing densities. Supplemental analyses, in addition to the DEM simulations, were used to develop a more comprehensive understanding of the system dynamics. The assembly stiffness and mass matrices were extracted from the DEM model and these data were used in an eigenmode analysis that provided significant insight into the observed overall dynamic response. The close agreement of the wave velocities estimated using eigenmode analysis with the DEM results confirms that DEM wave propagation simulations can reliably be used to extract material stiffness data. The data show that increasing either stress or density allows higher frequencies to propagate through the media, but the low-pass wavelength is a function of packing density rather than stress level. Prior research which had hypothesised that there is a simple link between the spectrum of the received signal and the natural sample frequencies was not substantiated.     

地震时桥梁碰撞分析的等效Kelvin撞击模型

建立地震时桥梁碰撞分析的Kelvin撞击模型的参数确定方法。基于Hertz接触理论,考虑波动效应,按照最大撞击力与最大撞击变形的比值,确定Kelvin撞击模型的碰撞刚度;数值分析了影响Kelvin撞击模型阻尼系数取值的邻梁碰撞恢复系数。结果表明:Kelvin撞击模型的碰撞刚度随Hertz接触刚度、撞击速度以及短梁与长梁的长度比的增大而增大;波动效应对碰撞刚度的影响明显,不能忽略;邻梁碰撞恢复系数随撞击速度增大而减小,随短梁与长梁的长度比减小而减小;确定Kelvin撞击模型的碰撞刚度和邻梁碰撞恢复系数对城市桥梁的合理取值范围应分别为3×105kN/m―6×105kN/m和0.7―0.95。     

Sensitivity of numerical parameters on DEM predictions of sediment transport

The current work presents a sensitivity study of selected numerical parameters on the large-eddy simulation–discrete element method predictions of sediment transport in a unidirectional open turbulent channel. The sensitivity of the particle friction factor, restitution coefficient, and spring stiffness used in the soft sphere collision model is tested in the three regimes of sediment transport comprising of essentially no motion, bed load, and suspended load flow regimes. The simulations are run for 10?s using base values of the parameters of interest (reference calculation), then one parameter is changed at a time and the corresponding change in quantitative result is observed for the following 15?s where averaged results in the last 10?s are compared to the reference calculation. The sensitivity analysis shows that the underpredicted sediment transport in the suspended load regime can be bridged by moderately decreasing the friction factor of the particles from 0.6 to 0.25. The impact of the same change in bed load regime is not as significant. Both the coefficient of restitution and the particle stiffness show less significant to negligible impact as compared to the friction factor.     

Micro-mechanical modelling of granular material. Part 2: Plane wave propagation in infinite media

Summary This paper discusses the propagation of plane body waves through a second-gradient micropolar elastic continuum. In an accompanying paper, this macroscopic constitutive law has been derived from the micro-level particle characteristics, which are the inter-particle stiffness, the particle size and the package density. As a result of incorporating the micro-scale effects, the body waves propagate in a dispersive manner, where dispersion becomes more prominent when the wavelength of the generated body waves reaches the order of magnitude of the particle size. After successively deriving the equations of motion and the dispersion relations for plane body wave propagation, the compressional wave properties for the second-gradient micro-polar model are compared to those for the Born-Karman lattice structure. Furthermore, distinguished features of the second-gradient micro-polar model are exhibited by comparing the dispersion relations of the coupled propagation of the shear wave and the micro-rotational wave with those of more simple constitutive models. The paper ends with a parameter study, where the effect by the translational particle contact stiffness and the rotational particle contact stiffness is examined.