Dispersion characteristics of wave propagation in layered piezoelectric structures: Exact and simplified models

This article presents a study of the dispersion characteristics of wave propagation in layered piezoelectric structures under plane strain and open-loop conditions. The exact dispersion relation is first determined based on an electro-elastodynamic analysis. The dispersion equation is complicated and can be solved only by numerical methods. Since the piezoelectric layer is very thin and can be modeled as an electro-elastic film, a simplified model of the piezoelectric layer reduces this complex problem to a non-trivial solution of a series of quadratic equations of wave numbers. The model is simple, yet captures the main phenomena of wave propagation. This model determines the dispersion curves of PZT4-Aluminum layered structures and identifies the two lowest modes of waves: the generalized longitudinal mode and the generalized Rayleigh mode. The model is validated by comparing with exact solutions, indicating that the results are accurate when the thickness of the layer is smaller or comparable to the typical wavelength. The effect of the piezoelectricity is examined, showing a significant influence on the generalized longitudinal wave but a very limited effect on the generalized Rayleigh wave. Typical examples are provided to illustrate the wave modes and the effects of layer thickness in the simplified model and the effects of the material combinations.     

Wave propagation through mangrove forests in the presence of a viscoelastic bed

The influence of viscoelastic ocean beds on the characteristics of surface waves passing through mangrove forests is analyzed under the assumption of linearized water wave theory in two dimensions. The trunks of the mangroves are assumed to be in the upper-layer inviscid fluid domain, whilst the roots are inside the viscoelastic bed. The associated equation of motion is obtained by coupling the Voigt’s model for flow within the viscoelastic medium with the equation of motion in the presence of mangroves. The modified dynamic conditions are coupled with the kinematic conditions to obtain the boundary condition at the free surface and the interface of the two fluids consisting of the upper layer inviscid fluid and the viscoelastic fluid bed. To understand the effects of bed viscosity as well as elasticity on energy dissipation, the complex dispersion relation associated with the plane progressive wave is derived and analyzed. Effect of physical parameters associated with mangroves and viscoelastic bed on wave motion in surface and internal modes are computed and analyzed to understand their roles in attenuating wave effects. The present model will be useful in the better understanding of wave propagation through mangroves in the coastal zone having muddy seabed.     

Shell element based model for wave propagation analysis in multi-wall carbon nanotubes

A spectrally formulated finite element is developed to study very high frequency elastic waves in carbon nanotubes (CNTs). A multi-walled nanotube (MWNT) is modelled as an assemblage of shell elements connected throughout their length by distributed springs, whose stiffness is governed by the van der Waals force acting between the nanotubes. The spectral element is formulated using the recently developed strategy based on singular value decomposition (SVD) and polynomial eigenvalue problem (PEP). The element can model a MWNT with any number of walls. Studies are carried out to investigate the effect of the number of walls on the spectrum and dispersion relation. The importance of shell element based model over the beam model is established. The zone of validity of the previously developed beam model is also investigated. It is shown that the shell model is required to capture the symmetric Lamb wave modes. It is also shown through numerical examples that the developed element efficiently captures the response of MWNT for Tera-hertz level frequency loading.     

SHEAR-HORIZONTAL WAVES IN A ROTATED Y-CUT QUARTZ PLATE WITH AN ISOTROPIC ELASTIC LAYER OF FINITE THICKNESS

We study shear-horizontal (SH) waves in a rotated Y-cut quartz plate carrying an isotropic elastic layer of finite thickness.The three-dimensional theories of anisotropic elasticity and isotropic elast...     

Internal wave reflections and transmissions arising from a non-uniform mesh. Part III: The occurrence of evanescent waves in the Crank-Nicolson linear finite element scheme

The numerical scheme upon which this paper is based is the 1D Crank–Nicolson linear finite element scheme. In Part I of this series it was shown that for a certain range of incident wavelengths impinging on the interface of an expansion in nodal spacing, an evanescent (or spatially damped) wave results in the downstream region. Here in Part III an analysis is carried out to predict the wavelength and the spatial rate of damping for this wave. The results of the analysis are verified quantitatively with seven ‘hot-start’ numerical experiments and qualitatively with seven ‘cold-start’ experiments. Weare has shown that evanescent waves occur whenever the frequency of a disturbance at a boundary exceeds the maximum frequency given by the dispersion relation. In these circumstances the ‘extended dispersion’ relation can be used to determine the rate of spatial decay. In the context of a domain consisting of two regions with different nodal spacings, the use of the group velocity concept shows that evanescent waves have no energy flux associated with them when energy is conserved.     

Overall constitutive description of symmetric layered media based on scattering of oblique SH waves

This papers investigates the scattering of oblique shear horizontal (SH) waves off finite periodic media made of elastic and viscoelastic layers. It further considers whether a Willis-type constitutive matrix (in temporal and spatial Fourier domain) may reproduce the scattering matrix (SM) of such a system. In answering this question the procedure to determine the relevant overall constitutive parameters for such a medium is presented. To do this, first the general form of the dispersion relation and impedances for oblique SH propagation in such coupled Willis-type media are developed. The band structure and scattering of layered media are calculated using the transfer matrix (TM) method. The dispersion relation may be derived based on the eigen-solutions of an infinite periodic domain. The wave impedances associated with the exterior surfaces of a finite thickness slab are extracted from the scattering of such a system. Based on reciprocity and available symmetries of the structure and each constituent layer, the general form of the dispersion and impedances may be simplified. The overall quantities may be extracted by equating the scattering data from TM with those expected from a Willis-type medium. It becomes evident that a Willis-type coupled constitutive tensor with components that are assumed independent of wave vector is unable to reproduce all oblique scattering data. Therefore, non-unique wave vector dependent formulations are introduced, whose SM matches that of the layered media exactly. It is further shown that the dependence of the overall constitutive tensors of such systems on the wave vector is not removable even at very small frequencies and incidence angles and that analytical considerations significantly limit the potential forms of the spatially dispersive constitutive tensors.     

Steady-state vibrations of an elastic beam on a visco-elastic layer under moving load

Summary  The steady-state response of an elastic beam on a visco-elastic layer to a uniformly moving constant load is investigated. As a method of investigation the concept of “equivalent stiffness” of the layer is used. According to this concept, the layer is replaced by a 1D continuous foundation with a complex stiffness, which depends on the frequency and the wave number of the bending waves in the beam. This stiffness is analyzed as a function of the phase velocity of the waves. It is shown that the real part of the stiffness decreases severely as the phase velocity tends to a critical value, a value determined by the lowest dispersion branch of the layer. As the phase velocity exceeds the critical value, the imaginary part of the equivalent stiffness grows substantially. The dispersion relation for bending waves in the beam is studied to analyze the effect of the layer depth on the critical (resonance) velocity of the load. It is shown that the critical velocity is in the order of the Rayleigh wave velocity. The smaller the layer depth, the higher the critical velocity. The effect of viscosity in the layer on the resonance vibrations is studied. It is shown that the deeper the layer, the smaller this effect. Received 22 March 1999; accepted 26 July 1999     

爆炸载荷下地下巷道分配层结构的对比分析

为寻找削波效果更好的分配层结构,基于光滑粒子流体动力学-有限元耦合方法SPH-FEM,建立地下巷道爆炸效应分析模型,形成11种不同的模拟方案。将含不同材料的分配层结构的测点应力峰值进行对比,以及从测点应力峰值、能量峰值和拱顶竖向位移的角度来比较不同分配层结构的削波能力,并将爆炸近区测点应力峰值与经验公式进行对比。结果表明,砂土的削波能力要强于泡沫混凝土;分配层中各层材料采用周期循环、波阻抗值递增且相邻材料波阻抗比减小的方式来布置,同时在分配层中加入钢铁,对应力波峰值的衰减和弥散效果最优。爆炸近区的模拟数值与计算值吻合度较高,证明此模拟具有一定的可靠性,可以为军事防护工程的结构设计提供参考。     

The Peierls stress in non-local elasticity

The effect of non-locality on the Peierls stress of a dislocation, predicted within the framework of the Peierls-Nabarro model, is investigated. Both the integral formulation of non-local elasticity and the gradient elasticity model are considered. A modification of the non-local kernel of the integral formulation is proposed and its effect on the dislocation core shape and size, and on the Peierls stress are discussed. The new kernel is longer ranged and physically meaningful, improving therefore upon the existing Gaussian-like non-locality kernels. As in the original Peierls-Nabarro model, lattice trapping cannot be captured in the purely continuum non-local formulation and therefore, a semi-discrete framework is used. The constitutive law of the elastic continuum and that of the glide plane are considered both local and non-local in separate models. The major effect is obtained upon rendering non-local the constitutive law of the continuum, while non-locality in the rebound force law of the glide plane has a marginal effect. The Peierls stress is seen to increase with increasing the intrinsic length scale of the non-local formulation, while the core size decreases accordingly. The solution becomes unstable at intrinsic length scales larger than a critical value. Modifications of the rebound force law entail significant changes in the core configuration and critical stress. The discussion provides insight into the issue of internal length scale selection in non-local elasticity models.     

Linear gravity waves on Maxwell fluids of finite depth

Linear surface gravity waves on Maxwell viscoelastic fluids with finite depth are studied in this paper. A dispersion equation describing the spatial decay of the gravity wave in finite depth is derived. A dimensionless memory (time) number 0 is introduced. The dispersion equation for the pure viscous fluid will be a specific case of the dispersion equation for the viscoelastic fluid as θ=0. The complex dispersion equation is numerically solved to investigate the dispersion relation. The influences of θ and water depth on the dispersion characteristics and wave decay are discussed. It is found that the role of elasticity for the Maxwell fluid is to make the surface gravity wave on the Maxwell fluid behave more like the surface gravity wave on the inviscid fluid.     

Higher order topological derivatives in elasticity

The topological derivative provides the variation of a response functional when an infinitesimal hole of a particular shape is introduced into the domain. In this work, we compute higher order topological derivatives for elasticity problems, so that we are able to obtain better estimates of the response when holes of finite sizes are introduced in the domain. A critical element of our algorithm involves the asymptotic approximation for the stress on the hole boundary when the hole size approaches zero; it consists of a composite expansion that is based on the responses of elasticity problems on the domain without the hole and on a domain consisting of a hole in an infinite space. We present a simple example in which the higher order topological derivatives of the total potential energy are obtained analytically and by using the proposed asymptotic expansion. We also use the finite element method to verify the topological asymptotic expansion when the analytical solution is unknown.     

冲击作用下含孔洞纤维增强复合材料平板的动力学实验与数值研究

作为复合材料动力学实验与数值研究的应用实例 ,实验研究采用正交异性动态光弹性方法 ,数值分析运用各向异性介质的时域边界元方法。纤维增强光弹性复合材料平板被用来模拟含孔洞的正交异性半无限域 ,用小口径步枪施加与纤维方向成 0及 90两个方向的冲击载荷 ,在正交异性动态光弹性实验中记录了应力波在孔洞周围的传播、反射与绕射过程 ,此过程被进一步转换成应力分量的变化时程 ,并与相应的时域边界元方法的数值分析结果进行了比较。     

Hydroelastic analysis of very large floating structure over viscoelastic bed

Effect of viscoelastic bed on the hydroelastic response analysis of very large floating structures is studied using the linear water wave theory and small amplitude structural response in finite water depth. The floating structure is modeled using Euler–Bernoulli beam equation and the bottom bed is assumed to be viscoelastic in nature and is based on the Voigt’s model. The dispersion relation, phase speed and response amplitude of the floating structure as well as viscoelastic bed surface, pressure distribution along water depth are analyzed to study the effect of viscoelastic bed parameters, flexural rigidity of the floating structure, time period on flexural gravity wave motion. The study reveals that structural response of the floating structure can be mitigated for moderate thickness of the viscoelastic layer. Moreover, both shear modulus and viscosity of the viscoelastic layer play dominant role in reducing the structural response compared to the flexural rigidity of the structure. Further, pressure distribution within the viscoelastic bed decreases at a higher rate compared to the inviscid fluid layer irrespective of shear modulus and viscosity. The present study will be of immense help in the site selection of very large floating structures in the coastal water and installation of various marine facilities over muddy bed.     

Internal waves in a two‐layer system using fully nonlinear internal‐wave equations

In order to understand the nonlinear effect in a two‐layer system, fully nonlinear strongly dispersive internal‐wave equations, based on a variational principle, were proposed in this study. A simple iteration method was used to solve the internal‐wave equations in order to solve the equations stably. The applicability of the proposed numerical computation scheme was confirmed to agree with linear dispersion relation theoretically obtained from variational principle. The proposed computational scheme was also shown to reproduce internal waves including higher‐order nonlinear effect from the analysis of internal solitary waves in a two‐layer system. Furthermore, for the second‐order numerical analysis, the balance of nonlinearity and dispersion was found to be similar to the balance assumed in the KdV theory and the Boussinesq‐type equations. Copyright © 2009 John Wiley & Sons, Ltd.     

应力脉冲在SHPB实验中弥散效应的数值模拟与频谱分析

在评判不同材料的整形器对加载波形的改进效果的应用背景下,对比研究了不同形态的应力脉冲在霍普金森压杆(split Hopkinson pressure bar,SHPB)中的弥散效应。利用有限元软件LS-DYNA建立了杆件的三维有限元模型,在杆端分别施加矩形、三角形和半正弦形的应力脉冲,分析了波形振荡、前沿升时和应力峰值随传播距离的变化规律,并运用频谱分析的方法进行了理论解释。结果表明:三角应力脉冲和半正弦应力脉冲在各个方面都比传统的矩形应力脉冲表现出了更小的弥散效应;半正弦应力脉冲在传播过程中比三角应力脉冲更能控制其形态,能有效地减少弥散效应,提高Φ100mm SHPB实验精度,是岩石类非均质材料的理想加载波形;频谱分析的方法能从理论方面有效地解释应力脉冲信号在SHPB实验中的弥散现象。由此可见,波形整形设计的理想目标为具有较宽历时的半正弦应力脉冲。     

Sparse element for shallow water wave equations on triangle meshes

Within the mixed FEM, the mini‐element that uses a bubble shape function for the solution of the shallow water wave equations on triangle meshes is simplified to a sparse element formulation. The new formulation has linear shape functions for water levels and constant shape functions for velocities inside each element. The suppression of decoupled spurious solutions is excellent with the new scheme. The linear dispersion relation of the new element has similar advantages as that of the wave equation scheme (generalised wave continuity scheme) proposed by Lynch and Gray. It is shown that the relation is monotonic over all wave numbers. In this paper, the time stepping scheme is included in the dispersion analysis. In case of a combined space–time staggering, the dispersion relation can be improved for the shortest waves. The sparse element is applied in the flow model Bubble that conserves mass exactly. At the same time, because of the limited number of degrees of freedom, the computational efficiency is high. The scheme is not restricted to orthogonal triangular meshes. Three test cases demonstrate the very good accuracy of the proposed scheme. The examples are the classical quarter annulus test case for the linearised shallow water equations, the hydraulic jump and the tide in the Elbe river mouth. Copyright © 2013 John Wiley & Sons, Ltd.     

功能梯度空心圆柱体中的周向超声导波

为了更有效地使用超声技术检测功能梯度管道中纵向缺陷,基于3维弹性理论,采用一种正交多项式法研究无限长功能梯度材料管道中的周向导波。为验证方法的正确性与适用性,首先采用该方法计算均匀各向异性管道中的周向导波,并与已有数据相比较。其次,计算了不同梯度函数下管道周向导波的频散曲线,并与均匀管道进行了比较,说明了梯度材料管道中周向导波的特点以及梯度函数变化对频散曲线的影响。最后,求解了不同径厚比下功能梯度材料圆柱壳周向导波的频散曲线,分析了径厚比的变化对频散曲线的影响。     

Orientational instability of a lyotropic nematic liquid crystal layer

The problem of periodic domain initiation in a thin lyotropic nematic liquid crystal layer is studied. This layer has a planar director initial orientation, but the anchoring energy is minimized by the homeotropic one. The periodic structures whose wave vector is perpendicular to the director exist during the director reorientation process from the planar orientation to the homeotropic one when the reorientation wave front appears. It is shown that the divergent terms of the Prank orientation elasticity energy plays an important role in this effect. The saddle-splay Prank constant and the anisotropic anchoring energy coefficient are estimated.     

Vertical momentum transfer by weakly nonlinear inertia-gravity internalwaves

Free inertia-gravity internal waves in a plane-parallel flow are considered in the Boussinesq approximation with account of the horizontal turbulent viscosity and diffusion. The dispersion relation and the wave decay rate are derived in the linear approximation. The effect of critical layers, in which the wave frequency with the Doppler shift is equal to the inertial frequency, on the dispersion curves is considered. It is shown that the dispersion curves are cut off in the low-frequency domain due to the critical layers mentioned above. The verticalwave momentum fluxes are nonzero and can be greater than the corresponding turbulent fluxes. It is shown that the Stokes drift velocity component transverse to the direction of propagation of the wave is nonzero with account of the turbulent viscosity and diffusion.     

多层波导中矢量波动的时域人工边界条件

本文提出了一种近似的时域人工边界条件(artificial boundary condition,ABC)用来模拟含有瑞利阻尼的线弹性多层波导平面内的矢量波动,该ABC是时域稳定的,且能与有限元法无缝耦合.建立ABC的思路是,首先将多层波导的矢量波动方程简化为x方向和y方向解耦的两个标量波动方程;其次基于比例边界有限元...