高度非线性孤立波与弹性大板的耦合作用研究

一维颗粒链的一端受到一个有初速度颗粒的撞击,导致颗粒连中产生稳定传播的应力波——高度非线性孤立波,该应力波的波长、波速以及幅值都能保持很好的稳定性,且遇到边界才会反射. 孤立波是一种良好的信息载体,广泛应用于无损检测技术中. 基于孤立波的特性,研究高度非线性孤立波与弹性大板耦合作用,基于赫兹定律和板的内在非弹性理论,推导出晶体链与大板的耦合微分方程组. 用龙格库塔法求解该微分方程组,得到颗粒链中各颗粒的位移、速度曲线. 通过分析回弹波出现的时间、回弹波所携带的能量以及模量、厚度、重力等对孤立波的影响,发现反射孤立波对大板的弹性模量和厚度尤为敏感,此外,颗粒链的摆放对整个耦合过程也有影响. 研究的结果为孤立波对结构体的无损探伤提供了理论依据,该技术可实现对结构体的快速检查和可控性研究.     

孤立波与半无限复合材料体耦合作用研究

基于一维颗粒链中产生的高度非线性孤立波,研究孤立波与半无限复合材料体的耦合作用。根据赫兹定律推导了一维颗粒链中颗粒间相互作用的运动微分方程,建立了颗粒链与半无限复合材料体的接触模型。对于颗粒与复合材料的接触,采用已有文献中修正后的赫兹定律,研究了高度非线性孤立波与半无限复合材料体的耦合力学作用机理,推导了颗粒链与半无限复合材料体的相互耦合运动微分方程组,通过数值计算,得到了各颗粒的内力、速度、位移曲线。分析了材料属性对回弹孤立波出现的时间、幅值的影响。结果表明:随着纤维方向弹性模量的增大,次级回弹波出现的时间和波幅都逐渐增大,随着垂直纤维方向弹性模量的增大,次级回弹波出现的时间先减小后增大,次级回弹波的幅值逐渐减小直至消失。     

Experimental Study of Strongly Nonlinear Resonances and Anti-Resonances in Granular Dimer Chains

Previous theoretical works considered the intrinsic dynamics of one-dimensional uncompressed granular dimer (diatomic) chains composed of pairs of dissimilar spherical elastic beads in Hertzian interaction. Such ordered granular media exhibit essentially nonlinear acoustics and have been characterized as ‘sonic vacua’ due to the fact that the speed of sound in these media (as defined in classical acoustics) is zero. Yet, depending on the mass ratios of the pairs of dissimilar beads of these dimers, it was proven that they may possess countable infinities of anti-resonances leading to solitary waves (this in spite of their high inhomogeneity), or countable infinities of strongly nonlinear resonances leading to passive strong attenuation of propagating pulses through energy radiation by means of excitation of traveling waves. The aim of this work is to experimentally verify the existence of these strongly nonlinear dynamics through a series of experiments involving granular dimer chains supported by flexures. By carefully designing the supporting flexures so that their dynamics is sufficiently ‘soft’ and thus separate from the ‘stiff’ dynamics governing the bead to bead interactions, we overcome a basic limitation for the experimental realization of such dimer systems, namely the construction of one-dimensional dimer chains with beads of different radii. Our results confirm experimentally the occurrence of nonlinear resonances and anti-resonances in dimer chains, and conclusively prove the capacity of appropriately designed granular dimers for passive strong attenuation of propagating pulses due to nonlinear resonance. Moreover, we validate the theoretical prediction that within the elastic range of bead to bead dynamical interactions the results are fully re-scalable with respect to energy. This work provides the first experimental evidence of strongly nonlinear resonances and anti-resonances in essentially nonlinear ordered granular media.     

On the coupling mechanism between nonlinear solitary waves and slender beams

This paper describes the coupling mechanism between highly nonlinear solitary waves propagating along a granular system and slender beams in contact with the granular medium. Nonlinear solitary waves are compact non-dispersive waves that can form and travel in nonlinear systems such as one-dimensional chains of particles, where they are conventionally generated by the mechanical impact of a striker. These waves have a constant spatial wavelength and their speed, amplitude, and duration can be tuned by modifying the particles’ material or size, or the velocity of the striker. In the study presented in this article we investigated numerically the interaction between solitary waves propagating along a chain of granular particles and a slender beam. Some of the numerical findings were validated experimentally. We found that the geometric and mechanical properties of the beam or thermal stress applied to the beam alter certain features of the solitary waves. In the future, these findings may be used to develop a novel sensing system for the nondestructive assessment of beams.     

Attenuation of Solitary Waves and Localization of Breathers in 1D Granular Crystals Visualized via High Speed Photography

We investigate the propagation, attenuation, and localization of nonlinear elastic waves in a 1D granular crystal using high speed photography. We measure temporal displacement profiles of individual particles with a micrometer-scale resolution, and we reconstruct force profiles of propagating solitary waves and localized breathers by synchronizing and analyzing the acquired data. These investigations provide quantitative evidence for the transmission and attenuation trends of travelling solitary waves in a soft polymeric chain, which are significantly different from those in a hard metallic chain. We additionally study energy localization in a chain of hard particles embedded with a soft polymeric impurity. Specifically, we show that the proposed experimental technique is able to visualize the formation of localized breathers and quantify the energy highly concentrated in the vicinity of the impurity site—a phenomenon which can be exploited for harvesting vibrational energy in engineering applications. Finally, we compare, with good agreement, the experimental results with discrete element numerical simulations that account for dissipative effects due to viscoelasticity. The findings reported in this study imply that high speed photography can be an efficient and effective tool for non-contact measurements of nonlinear wave dynamics in granular lattices, despite their short characteristic times and minute displacements.     

Frequency- and Amplitude-Dependent Transmission of Stress Waves in Curved One-Dimensional Granular Crystals Composed of Diatomic Particles

We study the stress wave propagation in curved chains of particles (granular crystals) confined by bent elastic guides. We report the frequency- and amplitude-dependent filtering of transmitted waves in relation to various impact conditions and geometrical configurations. The granular crystals studied consist of alternating cylindrical and spherical particles pre-compressed with variable static loads. First, we excite the granular crystals with small-amplitude, broadband perturbations using a piezoelectric actuator to generate oscillatory elastic waves. We find that the linear frequency spectrum of the transmitted waves creates pass- and stop-bands in agreement with the theoretical dispersion relation, demonstrating the frequency-dependent filtering of input excitations through the diatomic granular crystals. Next, we excite high-amplitude nonlinear pulses in the crystals using striker impacts. Experimental tests verify the formation and propagation of highly nonlinear solitary waves that exhibit amplitude-dependent attenuation. We show that the wave propagation can be easily tuned by manipulating the pre-compression imposed to the chain or by varying the initial curvature of the granular chains. We use a combined discrete element (DE) and finite element (FE) numerical model to simulate the propagation of both dispersive linear waves and compactly-supported highly nonlinear waves. We find that the tunable, frequency- and amplitude-dependent filtering of the incoming signals results from the close interplay between the granular particles and the soft elastic media. The findings in this study 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 frequencies and amplitudes.     

Solitary waves in a peridynamic elastic solid

The propagation of large amplitude nonlinear waves in a peridynamic solid is analyzed. With an elastic material model that hardens in compression, sufficiently large wave pulses propagate as solitary waves whose velocity can far exceed the linear wave speed. In spite of their large velocity and amplitude, these waves leave the material they pass through with no net change in velocity and stress. They are nondissipative and nondispersive, and they travel unchanged over large distances. An approximate solution for solitary waves is derived that reproduces the main features of these waves observed in computational simulations. It is demonstrated by numerical studies that the waves interact only weakly with each other when they collide. Wavetrains composed of many non-interacting solitary waves are found to form and propagate under certain boundary and initial conditions.     

Interaction of highly nonlinear solitary waves with thin plates

We investigate the reflection of highly nonlinear solitary waves in one-dimensional granular crystals interacting with large plates. We observe significant changes in the reflected waves’ properties in terms of wave amplitude and time of flight in association with the intrinsic inelasticity of large plates, which are governed by the plate thickness and the size of the granular constituents. We also study the effects of fixed plate boundaries in the formation of reflected waves, and find the existence of a critical distance, within which the interaction between the granular chain and plate is strongly modified. We explain the effects of intrinsic inelasticity and of boundaries in the large plates by using plate theory and the contact mechanics between a plate and a spherical striker. We find that experimental results are in excellent agreement with the analytical predictions and numerical simulations based on the combined discrete element and spectral element models. The findings in this study can be useful for the nondestructive evaluation of plate structures using granular crystals, which can improve the resolution of in-situ, portable measurement instruments leveraging high acoustic energy and sensitivity of solitary waves.     

Directional Wave Propagation in a Highly Nonlinear Square Packing of Spheres

We studied the dynamic response of a two-dimensional square packing of uncompressed stainless steel spheres excited by impulsive loadings. We developed a new experimental measurement technique, employing miniature tri-axial accelerometers, to determine the stress wave properties in the array resulting from both an in-plane and out-of-plane impact. Results from our numerical simulations, based on a discrete particle model, were in good agreement with the experimental results. We observed that the impulsive excitations were resolved into solitary waves traveling only through initially excited chains. The observed solitary waves were determined to have similar (Hertzian) properties to the extensively studied solitary waves supported by an uncompressed, uniform, one-dimensional chain of spheres. The highly directional response of this system could be used as a basis to design granular crystals with predetermined wave propagation paths capable of mitigating stress wave energy.     

INTRACELLULAR SOLITARY PULSE CALCIUM WAVES IN FROG SYMPATHETIC NEURONS

IntroductionCalciumionlevelsinintracellularsympatheticneuronsplayacontrollingroleinthetransmissionprocessesofvarioussignals,includingneuron_transmittersecretion ,regulationofmembraneexcitability ,inductionofgeneexpression ,etc .[1]However,howisthecytoplasmiccalciumconcentrationinneuronsregulated ?WehavealreadyknownthattheraiseincalciumconcentrationinXenopuslaevisoocyteismainlyoriginatedfromtheinfluxofCa2 outofcalciumstoresintheendoplasmicreticulum ,whichisinitiatedwiththestimulationofextrace…     

Primary wave transmission in systems of elastic rods with granular interfaces

We study analytically and numerically primary pulse transmission in one dimensional systems of identical linearly elastic non-dispersive rods separated by identical homogeneous granular layers composed of n beads. The beads interact elastically through a strongly (essentially) nonlinear Hertzian contact law. The main challenge in studying pulse transmission in such strongly nonlinear media is to analyze the ‘basic problem’, namely, the dynamical response of a single intermediate granular layer, confined from both ends by barely touching linear elastic rods subject to impulsive excitation of the left rod. The analysis of the basic problem is carried out under two basic assumptions; namely, of sufficiently small duration of the shock excitation applied to the first layer of the system, and of sufficiently small mass of each bead in the granular interface compared to the mass of each rod. In fact, the smallness of the mass of the bead defines the small parameter in the asymptotic analysis of this problem. Both assumptions are reasonable from the point of view of practical applications. In the analysis we focus only in primary pulse propagation, by neglecting secondary pulse reflections caused by wave scattering at each granular interface and considering only the transmission of the main (primary) pulse across the interface to the neighboring elastic rod. Two types of shock excitations are considered. The first corresponds to fixed time duration (but still much smaller compared to the characteristic time of pulse propagation through the length of each rod), whereas the second type corresponds to a pulse duration that depends on the small parameter of the problem. The influence of the number of beads of the granular interface on the primary wave transmission is studied, and it is shown that at granular interfaces with a relatively low number of beads fast time scale oscillations are excited with increasing amplitudes with increasing number of beads. For a larger number of beads, primary pulse transmission is by means of solitary wave trains resulting from the dispersion of the original shock pulse; in that case fast oscillations result due to interference phenomena caused by the scattering of the main pulse at the boundary of the interface. Considering a periodic system of rods we demonstrate significant reduction of the primary pulse when transmitted through a sequence of granular interfaces. This result highlights the efficacy of applying granular interfaces for passive shock mitigation in layered elastic media.     

Propagation of plane waves at the imperfect boundary of elastic and electro-microstretch generalized thermoelastic solids

The problem of reflection and transmission of plane waves incident on the contact surface of an elastic solid and an electro-microstretch generalized thermoelastic solid is discussed. It is found that there exist five reflected waves, i.e., longitudinal displacement （LD） wave, thermal （T） wave, longitudinal microstretch （LM） wave and two coupled transverse displacement and microrotational （CD（I） and CD（II）） waves in the electro-microstretch generalized thermoelastic solid, and two transmitted waves, i.e., longitudinal （P） and transverse （SV） waves in the elastic solid. The amplitude ratios of different reflected and transmitted waves are obtained for an imperfect boundary and deduced for normal force stiffness, transverse force stiffness, and perfect bonding. The variations of amplitude ratios with incidence angles have been depicted graphically for the LD wave and the CD（I） wave. It is noticed that the amplitude ratios of reflected and transmitted waves are affected by the stiffness, electric field, stretch, and thermal properties of the media. Some particular interest cases have been deduced from the present investigations.     

Non-linear waves in a viscous fluid contained in an elastic tube with variable cross-section

In the present work, treating the large arteries as a thin-walled, long and circularly cylindrical, prestressed elastic tube with variable cross-section and using the reductive perturbation method, we have studied the amplitude modulation of non-linear waves in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, the evolution equation is obtained as the dissipative non-linear Schrödinger equation with variable coefficients. It is shown that this type of equations admit a solitary wave solution with a variable wave speed. It is observed that, the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

INTERNAL SOLITARY WAVE LOADS EXPERIMENTS AND ITS THEORETICAL MODEL FOR A CYLINDRICAL STRUCTURE

在大型重力式密度分层水槽中, 对内孤立波与圆柱型结构的相互作用特性开展了系列实验. 基于两层流体中 内孤立波的KdV,eKdV和MCC理论, 建立了圆柱型结构内孤立波载荷的理论预报模型, 给出了该载荷理论预报模型中3类内孤立波理论的适用性条件.研究表明, 圆柱型结构内孤立波水平载荷包括水平Froude-Krylov力、附加质量力和拖曳力3个部分, 可以由Morison公式计算, 而内孤立波垂向载荷主要为垂向Froude-Krylov力, 可以由内孤立波诱导动压力计算.系列实验结果表明, 附加质量系数可以取为常数1.0, 拖曳力系数与内孤立波诱导速度场的雷诺数之间为指数函数关系, 而且基于理论预报模型的数值结果与系列实验结果吻合.     

The evolution and interaction of Marangoni-Bénard solitary waves

Marangoni-Bénard convection is the process by which oscillatory waves are generated on an interface due to a change in surface tension. This process, which can be mass or temperature driven is described by a perturbed Korteweg-de Vries (KdV) equation. The evolution and interaction of solitary waves generated by Marangoni-Bénard convection is examined. The solitary wave with steady-state amplitude, which occurs when the excitation and friction terms of the perturbed KdV equation are in balance is found to second-order in the perturbation parameter. This solitary wave has a fixed amplitude, which depends on the coefficients of the perturbation terms in the governing equation. The evolution of a solitary wave of arbitrary amplitude to the steady-state amplitude is also found, to first-order in the perturbation parameter. In addition, by using a perturbation method based on inverse scattering, it is shown that the interaction of two solitary waves is not elastic with the change in wave amplitude determined. Numerical solutions of the perturbed KdV equation are presented and compared to the asymptotic solutions.     

基于微形态模型的颗粒材料中波的频散现象研究

基于颗粒材料冲击与波动响应特性的调控波传播行为的超材料设计受到广泛关注,设计这类材料需要对颗粒材料的波传播机制及调控机理有深入认识. 波在颗粒材料中传播的频散现象及频率带隙等行为与材料的非均匀性密切相关,通常讨论频散现象是基于弹性理论框架建立微结构连续体或高阶梯度连续体等广义连续体模型来进行. 本研究基于细观力学给出了一个颗粒材料的微形态连续体模型. 在该模型中,考虑了颗粒的平动和转动,且颗粒间的相对运动分解为两部分:即宏观平均运动和细观真实运动. 基于此分解,提出了一个完备的变形模式,得到了对应于不同应变及颗粒间运动的宏细观本构关系. 结合宏观变形能的细观变形能求和表达式,获得了基于细观量表示的宏观本构模量. 应用所建议模型考察了波在弹性颗粒介质的传播行为,给出了不同形式的波的频散曲线,结果显示此模型具有预测频率带隙的能力.      

Destabilization of long-wavelength Love and Stoneley waves in slow sliding

Love waves are dispersive interfacial waves that are a mode of response for anti-plane motions of an elastic layer bonded to an elastic half-space. Similarly, Stoneley waves are interfacial waves in bonded contact of dissimilar elastic half-spaces, when the displacements are in the plane of the solids. It is shown that in slow sliding, long-wavelength Love and Stoneley waves are destabilized by friction. Friction is assumed to have a positive instantaneous logarithmic dependence on slip rate and a logarithmic rate weakening behavior at steady-state.Long-wavelength instabilities occur generically in sliding with rate- and state-dependent friction, even when an interfacial wave does not exist. For slip at low rates, such instabilities are quasi-static in nature, i.e., the phase velocity is negligibly small in comparison to a shear wave speed. The existence of an interfacial wave in bonded contact permits an instability to propagate with a speed of the order of a shear wave speed even in slow sliding, indicating that the quasi-static approximation is not valid in such problems.     

Correlation between theoretical and experimental solitary waves

Experimental data on surface solitary waves generated by five methods are given. These data and literature information show that at amplitudes 0.2<a/h<0.6 (h is the initial depth of the liquid), experimental solitary waves are in good agreement with their theoretical analogs obtained using the complete model of liquid potential flow. Some discrepancy is observed in the range of small amplitudes. The reasons why free solitary waves of theoretically limiting amplitude have not been realized in experiments are discussed, and an example of a forced wave of nearly limiting amplitude is given. The previously established fact that during evolution from the state of rest, undular waves break when the propagation speed of their leading front reaches the limiting speed of propagation of a solitary wave is confirmed. Lavrent’ev Institute of Hydrodynamics, Siberian Division, Russian Academy of Sciences, Novosibirsk 630090. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 40, No. 3, pp. 44–52, May–June, 1999.     

基于势流理论的内孤立波与立管作用数值研究

内孤立波是一种发生在水面以下的在世界各个海域广泛存在的大幅波浪, 其剧烈的波面起伏所携带的巨大能量对以海洋立管为代表的海洋结构物产生严重威胁, 分析其传播演化过程的流场特征及立管在内孤立波作用下的动力响应规律对于海洋立管的设计具有重要意义. 本文基于分层流体的非线性势流理论, 采用高效率的多域边界单元法, 建立了内孤立波流场分析计算的数值模型, 可以实时获得内孤立波的流场特征. 根据获得的流场信息, 采用莫里森方程计算内孤立波对海洋立管作用的载荷分布. 将内孤立波流场非线性势流计算模型与动力学有限元模型结合来求解内孤立波作用下海洋立管的动力响应特征, 讨论了内孤立波参数、顶张力大小以及内部流体密度对立管动力响应的影响. 发现随着内孤立波波幅的增大, 海洋立管的流向位移和应力明显增大. 由于上层流体速度明显大于下层, 且在所研究问题中拖曳力远大于惯性力, 因此管道顺流向的最大位移发生在上层区域. 顶张力通过改变几何刚度阵的值进而对立管的响应产生明显影响. 对于弱约束立管, 内部流体的密度对管道的流向位移影响较小.      

Primary pulse transmission in coupled steel granular chains embedded in PDMS matrix: Experiment and modeling

We present an experimental study of primary pulse transmission in coupled ordered steel granular chains embedded in poly-di-methyl-siloxane (PDMS) elastic matrix. Two granular one-dimensional chains are considered (an ‘excited’ and an ‘absorbing’ one), each composed of 11 identical steel beads of 9.5 mm diameter with the centerline of the chain spaced at fixed distances of 0.5, 1.5 or 2.5 mm apart. We directly force one of the chains (the excited one) by a transient pulse and measure, by means of laser vibrometry, the primary transmitted pulses at the end beads of both chains and at the first bead of the absorbing chain. It is well known that the dynamics of this type of ordered granular media is strongly nonlinear due, (i) to Hertzian interactions between adjacent beads, and (ii) to possible bead separations in the absence of compressive forces and ensuing collisions between neighboring beads. Accordingly, we develop a strongly nonlinear theoretical model that takes into account the coupling of the granular chains due to the PDMS matrix, with the aim to model primary pulse transmission in this system. After validating the model with experimental measurements, we employ it in a predictive fashion to estimate energy transfer between chains as a function of the interspatial distance between chains. Furthermore, based on this model we perform predictive matrix design to achieve maximum energy transfer from the excited to the absorbing chain, and provide a theoretical explanation of the nonlinear dynamics governing energy transfer (including energy equi-partition) in this system.