WAVE PROPAGATION IN A SYSTEM OF COAXIAL TUBES FILLED WITH INCOMPRESSIBLE MEDIA: A MODEL OF PULSE TRANSMISSION IN THE INTRACRANIAL ARTERIES

The propagation of harmonic waves through a system formed of coaxial tubes filled with incompressible continua is considered as a model of arterial pulse propagation in the craniospinal cavity. The inner tube represents a blood vessel and is modelled as a thin-walled membrane shell. The outer tube is assumed to be rigid to account for the constraint imposed on the vessels by the skull and the vertebrae. We consider two models: in the first model the annulus between the tubes is filled with fluid; in the second model the annulus is filled with a viscoelastic solid separated from the tubes by thin layers of fluid. In both models, the elastic tube is filled with fluid. The motion of the fluid is described by the linearized form of the Navier–Stokes equations, and the motion of the solid by classical elasticity theory. The results show that the wave speed in the system is lower than that for a fluid-filled elastic tube free of any constraint. This is due to the stresses generated to satisfy the condition that the volume in the system has to be conserved. However, the effect of the constraint weakens as the radius of the outer tube is increased, and it should be insignificant for the typical physiological parameter range.     

On the evolution equation of longitudinal shock waves in elastic media with weak inhomogeneity

Several problems of shock deformation in a nonlinearly elastic compressible medium with inhomogeneous properties are considered. The method of matched asymptotic expansions is used to show that the weak inhomogeneity and a certain relation between its order and the model nonlinearity order lead to different types of evolution quasilinear wave equations in regions far from the loaded boundary. The most interesting version of the arising evolution equation was obtained by joint change of the spatial coordinate scale and the related type of the semicharacteristic variable. The solution ideas are illustrated by an example of plane longitudinal shock wave in a medium with inhomogeneity in the wave motion direction. The obtained evolution equations become the well-known Cole-Hopf equation in the limit when passing to the isotropic medium.     

Mathematical model of spiral waves propagating in the aorta

The spiral waves in the viscous incompressible fluid flow within an arterial vessel modeled by a thin elastic isotropic shell are studied. Asymptotic expansions are constructed for two types of spiral waves. The first type is spiral long wall waves generated (owing to the viscous fluid no-slip at the inner shell wall) by the longitudinal and twist harmonic waves that propagate along the wall. For these waves the amplitude distribution over the vessel cross-section has the form of a boundary layer localized near the inner shell surface. The second is short small-amplitude waves that practically fill the entire vessel cross-section. It is shown that for the short waves the transfer mechanismis the steady-state flow, the role of the longitudinal wall waves and the elastic characteristics of the shell being in this case insignificant.     

Effect of small size on dispersion characteristics of wave in carbon nanotubes

Effect of small size on dispersion characteristics of waves in multi-walled carbon nanotubes is investigated using an elastic shell model. Dynamic governing equations of the carbon nanotube are formulated on the basis of nonlocal elastic theory. The relationship between wavenumber and frequency of wave propagation is obtained from the solution of the eigenvalue equations. The numerical results show that the dispersion characteristics of wave in the multi-walled carbon nanotube are affected by the small size. Effect of small size is not obvious for the smaller wavenumber, and it will arise and increase gradually with the increase of the wavenumber. Effect of the small size will decrease as the inner radius of carbon nanotubes increases. In addition, the explicit expressions of the cut-off frequencies are derived. The results show that the cut-off frequencies cannot be influenced by the small size of carbon nanotubes.     

非饱和土-隧道系统动力响应计算的波函数法

针对非饱和地基土中埋置隧道的三维动力响应计算问题, 提出了波函数法.采用无限长的Flügge薄壁圆柱壳模拟圆形隧道衬砌,采用流、固、气组成的三相介质模拟非饱和地基土体.分别采用分离变量法以及Helmholtz矢量分解定理求解薄壁圆柱壳的振动控制方程与非饱和土的波动方程.根据隧-土交界面与地表面处的应力、位移以及孔隙流体压力等边界条件,利用平面波与柱面波的转换性质,实现了隧道内作用单位简谐载荷时隧道衬砌与土体系统动力响应的耦合求解.通过与既有单相弹性介质2.5维有限元-边界元法、两相饱和多孔介质2.5维有限元-边界元法以及三相非饱和介质Pip in Pip半解析法的计算结果进行对比, 验证了本文计算方法的可靠性. 最后,基于该方法, 通过算例分析了不同饱和度下非饱和土-隧道系统的动力响应特征.结果表明, 饱和度对土体动位移与超孔隙水压力的幅值响应有较大影响.该方法的非饱和地基土参数退化后,也可用来计算和分析饱和地基土或单相弹性地基土与隧道系统的动力响应.      

The propagation of elastic stress waves in a conical shell under axial impulsive loading

The propagation of elastic stress waves in a conical shell subjected to axial impulsive loading is studied in this paper by means of the finite element calculation and model experiments. It is shown that there are two axisymmetrical elastic stress waves propagating with different velocities, i.e., the longitudinal wave and the bending wave. The attenuation of these waves while propagating along the shell surface is discussed. It is found in experiments that the bending wave is also generated when a longitudinal wave reflects from the fixed end of the shell, and both reflected waves will separate during the propagation due to their different velocities. Southwest Institute of Structural Mechanics     

The application of the Lattice Boltzmann method to the one-dimensional modeling of pulse waves in elastic vessels

The one-dimensional nonlinear equations for the blood flow motion in distensible vessels are considered using the kinetic approach. It is shown that the Lattice Boltzmann (LB) model for non-ideal gas is asymptotically equivalent to the blood flow equations for compliant vessels at the limit of low Knudsen numbers. The equations of state for non-ideal gas are transformed to the pressure-luminal area response. This property allows to model arbitrary pressure-luminal area relations. Several test problems are considered: the propagation of a sole nonlinear wave in an elastic vessel, the propagation of a pulse wave in a vessel with varying mechanical properties (artery stiffening) and in an artery bifurcation, in the last problem Resistor–Capacitor–Resistor (RCR) boundary conditions are considered. The comparison with the previous results shows a good precision.     

血管位移波的力学模型

本文在临床观察和模型实验的基础上提出了血管位移波的力学模型,并归结出了位移波的控制方程。应用摄动理论和数值方法,求得了位移波的波形和位移波产生的判别准则。从中我们发现,位移波仅和血流的脉动速度及血管壁力学特性有关,而和压力波无直接关系。最后本文还给出了一个适用于临床的位移波发生判别准则,实验发现它和临床观察是相符合的。     

Displacement wave of the blood vessel —A mechanical model

According to clinical observations and model experiment, we have developed a theoretical model and obtained its mathematical equations to investigate the displacement wave of the blood vessel. The waveform and its occurrence criterion have been obtained by using perturbation theory and numerical method. The present study suggests that the displacement wave is associated with the blood velocity waveform and the machanical behaviour of the blood vessel, but not with the pressure waveform. The clinical criterion is in agreement with the observations.     

Nonlinear dynamics of oriented elastic solids

Based on a continuum model for oriented elastic solids the set of nonlinear dispersive equations derived in Part I of this work allows one to investigate the nonlinear wave propagation of the soliton type. The equations govern the coupled rotation-displacement motions in connection with the linear elastic behavior and large-amplitude rotations of the director field. In the one-dimensional version of the equations and for two simple configurations an exhaustive study of solitons is presented. We show that the transverse and/or longitudinal elastic displacements are coupled to the rotational motion so that solitons, jointly in the rotation of the director and the elastic deformations, are exhibited. These solitons are solutions of a system of linear wave equations for the elastic displacements which are nonlinearly coupled to a sine-Gordon equation for the rotational motion. For each configuration, the solutions are numerically illustrated and the energy of the solitions is calculated. Finally, some applications of the continuum model and the related nonlinear dynamics to several physical situations are given and additional more complex problems are also evoked by way of conclusion.     

On stability of orthotropic cylindrical shells

In contrast to [1–3], the present paper obtains a system of stability equations and the corresponding resolving equation for orthotropic cylindrical shells of any but very short length in the case where the precritical stress state cannot be treated as the zero-moment state. These equations are a generalization of the results obtained in [4]. On the basis of these equations, one can obtain both the well-known formulas [1–3] and, for medium-length shells, some new expressions of the critical load in longitudinal compression and that under the joint action of torsionalmoments, normal pressure, and longitudinal compression. Some estimates are performed and the determination of the domain of application of some formulas given in [2] and in the present paper is attempted. For an orthotropic shell, a relationship between the elastic parameters and the shear modulus is established for axisymmetric and nonaxisymmetric buckling mode shapes in longitudinal compression.     

舰船海底地震波形成机理的理论分析

航行舰船是一个很大的振动和噪声源,其振动和噪声引起的压力波经由流体介质传播到海底,引起海底岩层振动,产生弹性波并向远处传播,这种航行舰船在海底引起的弹性波称为舰船海底地震波。舰船海底地震波可用于识别舰船目标,作为水中武器的目标探测信号。基于波动理论,得到了舰船引起海底地震波的波动方程和有限元数值计算模型。利用该模型,对二维半无限空间层状海洋环境在水压场和极低频点声源作用下的海底地震波信号进行了数值模拟,得到了远场海底表面处加速度和水压的时域频域信号。结果表明,在二维半无限空间层状海洋环境条件下,舰船远场海底地震波主要由舰船极低频的噪声辐射引起,主要频率范围在1-7Hz。     

Performance modeling and analysis of blood flow in elastic arteries

Nomenclature　　τ　　wallshearstressγshearrateτy yieldstressηc Cassonviscosityktheconsistencyindexnnon_Newtonianindexτp shearstressofthepthelementωangularvelocityRvessel’sradiusCwavespeedM　　magneticparameter (Hartmannnumber)u,w　velocitycomponentinther_andz_directions,respectivelyP　　pressureα　　unsteadinessparameter k , R meanparametersTp relaxationtimeofthepthelementρ densityIntroductionTheimportancetoatherogenesisofarterialflowphenomenasuchasflowseparation ,recirculationands…     

Effective equations describing the flow of a viscous incompressible fluid through a long elastic tubeEquations efficaces décrivant l'écoulement d'un fluide visqueux incompressible dans un tuyau long et élastique

We study the flow of a viscous incompressible fluid through a long and narrow elastic tube whose walls are modeled by the Navier equations for a curved, linearly elastic membrane. The flow is governed by a given small time dependent pressure drop between the inlet and the outlet boundary, giving rise to creeping flow modeled by the Stokes equations. By employing asymptotic analysis in thin, elastic, domains we obtain the reduced equations which correspond to a Biot type viscoelastic equation for the effective pressure and the effective displacement. The approximation is rigorously justified by obtaining the error estimates for the velocity, pressure and displacement. Applications of the model problem include blood flow in small arteries. We recover the well-known Law of Laplace and provide a new, improved model when shear modulus of the vessel wall is not negligible. To cite this article: S. ?ani?, A. Mikeli?, C. R. Mecanique 330 (2002) 661–666.     

轴向冲击载荷作用下锥壳中弹性应力波传播的计算和实验研究

本文利用有限元分析和模型实验研究了在轴向冲击载荷作用下,锥壳中弹性应力波的传播、计算和实验结果表明,结构中存在着弹性纵波和弹性弯曲波的传播,它们传播的速度各不相同,使壳面承受不同的应力状态;讨论了纵波和弯曲波随壳面的衰减;实验指出,由于边界的影响,即使纵波的反射也会产生新的反射弯曲波沿锥面传播。     

Wave propagation in carbon nanotubes embedded in an elastic matrix

This paper reports the results of an investigation into the characteristics of wave propagation in carbon nanotubes embedded in an elastic matrix, based on an exact shell model. Each of the concentric tubes of multi-walled carbon nanotubes is considered as an individual elastic shell and coupled together through the van der Waals forces between two adjacent tubes. The matrix surrounding carbon nanotubes is described as a spring element defined by the Winkler model. The effects of rotatory inertia and elastic matrix on the wave velocity, the critical frequency, and the amplitude ratio between two adjacent tubes are described and discussed through numerical examples. The results obtained show that wave propagation in carbon nanotubes may appear in a critical frequency at which the wave velocity changes suddenly; the elastic matrix surrounding carbon nanotubes debases the critical frequency and the wave velocity, and changes the vibration modes between two adjacent tubes; the rotatory inertia based on an exact shell model obviously influences the wave velocity at some wave modes. Finally, a comparison of dispersion solutions from different shell models is given. The present work may serve as a useful reference for the application and the design of nano-electronic and nano-drive devices, nano-oscillators, and nano-sensors, in which carbon nanotubes act as basic elements.     

球壳结构馈通增长的瑞利-泰勒不稳定性

利用小扰动分析法 ,导出不可压缩球壳结构的馈通增长方程 ,数值模拟了高压气体驱动外表面有初始扰动的明胶球壳的瑞利 泰勒不稳定性模型。计算结果表明 :对于低波数扰动 ,外界面比较稳定 ,内表面的馈通增长较快 ,具有比较明显的三个演化阶段和波形反转现象。高波数扰动的增长恰好与低波数相反。球壳会聚结构比柱壳会聚结构的界面稳定性要好些。     

爆炸容器内冲击波系演化及壳体响应的数值研究

对中心装药爆炸后冲击波的产生、传播和壳体动态响应全过程进行了数值研究。认为RDX瞬时爆炸 ,爆炸近场采用自相似解 ;冲击波传播和波系演化采用PPM (the Piecewise Parabolic Method)格式求解Eu ler方程 ;壳体响应采用有限元方法求解拉氏坐标系下由虚功原理得到的动力学方程。壳体内壁面边界条件分别采用强耦合和弱耦合方法处理。结果表明 :(1)当装药量相同时 ,薄壁壳体振型比厚壁壳体复杂得多 ,振幅也大 ;(2 )当装药量不同 ,壳体厚度相同时 ,爆炸场冲击波的演化过程不同 ;(3)对少量装药 ,产生的冲击波强度低 ,壳体变形小 ,是否考虑内边界运动 ,对计算结果的影响不大 ;(4 )在本文条件下 ,爆炸容器封头顶点所受的载荷最大 ,是最易发生破坏的地方 ,侧壁与爆点所在横截面的交线 ,也易破坏。     

Asymptotic Solutions of the Equations for a Viscous Heat-Conducting Compressible Medium

Small nonstationary perturbations in a viscous heat-conducting compressible medium are analyzed on the basis of the linearization of the complete system of hydrodynamic equations for small Knudsen numbers (Kn ≪ 1). It is shown that the density and temperature perturbations (elastic perturbations) satisfy the same wave equation which is an asymptotic limit of the hydrodynamic equations far from the inhomogeneity regions of the medium (rigid, elastic or fluid boundaries) as M _a = v/a → 0, where v is the perturbed velocity and a is the adiabatic speed of sound. The solutions of the new equation satisfy the first and second laws of thermodynamics and are valid up to the frequencies determined by the applicability limits of continuum models. Fundamental solutions of the equation are obtained and analyzed. The boundary conditions are formulated and the problem of the interaction of a spherical elastic harmonic wave with an infinite flat surface is solved. Important physical effects which cannot be described within the framework of the ideal fluid model are discussed.__________Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, 2005, pp. 76–87.Original Russian Text Copyright © 2005 by Stolyarov.