Nonlinear random stability of viscoelastic cable with small curvature

The nonlinear planar mean square response and the random stability of a viscoelastic cable that has a small curvature and subjects to planar narrow band random excitation is studied. The Kelvin viscoelastic constitutive model is chosen to describe the viscoelastic property of the cable material. A mathematical model that describes the nonlinear planar response of a viscoelastic cable with small equilibrium curvature is presented first. And then a method of investigating the mean square response and the almost sure asymptotic stability of the response solution is presented and regions of instability are charted. Finally , the almost sure asymptotic stability condition of a viscoelastic cable with small curvature under narrow band excitation is obtained.     

Some remarks on the growth behavior of one-dimensional shock waves in non-linear materials

The growth behavior of both compressive and expansive one-dimensional shock waves which propagate into an unstrained region of a non-linear material exhibiting anelastic response, in the sense of Eckart, is analyzed. In each case, a differential equation governing the growth of the amplitude of the shock is derived and it is shown that a critical strain gradient may be defined. The growth behavior of the waves closely resembles the growth behavior of compressive and expansive shock waves propagating in sufficiently smooth non-linear materials with fading memory, i.e., in materials which can be approximated by linear viscoelastic materials for small relative strains.     

Time–transient response for ultrasonic guided waves propagating in damped cylinders

Herein, an enhanced spectral finite element (SFE) formulation to calculate the time–transient response in cylindrical waveguides is proposed. The original aspect over SFE-based formulations consists in the possibility to account for the effect of material absorption, i.e. guided waves attenuation, on the calculation of the time–transient response.First, the damped steady-state response is constructed by a weighted superposition of the waveguide modal properties obtained from the spectral decomposition of the governing wave equation. To this purpose an enhanced spectrally formulated finite element is developed, in which material damping is included allowing for complex stress–strain viscoelastic constitutive relations in force of the correspondence principle. Dispersive modal properties for the damped waveguide (phase velocity, energy velocity, attenuation and wavestructures) follow straightforwardly by simple formulae. Next, the frequency response of the problem is calculated by weighting the modal data and the spectrum of the applied time-dependent force via Cauchy residue theorem. Finally, the inverse Fourier transform of the frequency response leads to the time–transient response for propagative damped guided waves.The approach is not restricted to any anisotropy degree, holds for any linear viscoelastic constitutive relation that can be characterized and formulated in the frequency domain and it can be applied to SFE formulations for arbitrary cross-section waveguides. A study on guided waves propagating in a scheduled 4.in-40 ANSI steel pipe is presented, where the steel is considered first as perfectly elastic and then as an hysteretic viscoelastic medium, in order to show the effect of material absorption on the time–transient response.     

Dispersion of linear gravity waves on a viscoelastic fluid in an horizontal canal

A characteristic equation is derived that describes the spatial decay of linear surface gravity waves on Maxwell fluids. Except at small frequencies, the derived dispersion relation is different from the temporal decay dispersion relation which is normally studied within fluid mechanics. The implications for waves on viscous Newtonian fluids using the two different dispersion relations is briefly discussed. The wave number is measured experimentally as function of the frequency in a horizontal canal. Seven Newtonian fluids and four viscoelastic liquids with constant viscosity have been used in the experiments. The spatial decay theory for Newtonian fluids fits well to the experimental data. The model and experiments are used to determine limits for the Maxwell fluid time numbers for the four viscoelastic liquids. As a result of low viscosity it was not possible within this study to obtain these time numbers from oscillatory experiments. Therefore, a comparison of surface gravity wave experiments with theory is applicable as a method to evaluate memory times of low viscosity viscoelastic fluids.     

Rayleigh wave propagation in curved waveguides

A JWKB asymptotic expansion describing inplane elastic waves is used to approximate a Rayleigh-like wave guided within a curved elastic waveguide whose curvature is small and changes slowly over a wavelength. The two lowest eigenmodes in a curved guide, taken together, constitute the Rayleigh-like wave. It is shown that this wave lies in the shadows of four, closely spaced, virtual caustics, two caustics per constituent eigenmode. If the curvature becomes too large one or more of the caustics ceases to be virtual and enters the guide after which a Rayleigh-like wave cannot propagate. The overall disturbance is shown to have an amplitude that is modulated because the wavenumbers of the constituent eigenmodes differ by a small amount. Moreover, the disturbance is shown to propagate with a wavenumber that, to leading order, has a linear dependence on the curvature causing the phase to be modulated, as well. Passing from a thin guide to a very thick one suppresses the amplitude modulation, making the phase modulation evident. Propagation into an environment of increasing curvature, for both thin and thick, shallowly curved guides is studied so that the modulations may be observed.     

The influence of constraints on the properties of acceleration waves in isotropic thermoelastic media

The properties of acceleration waves are investigated for situations in which the waves propagate in isotropic heat-conducting elastic media subject to arbitrary sets of constraints. Conditions under which waves may exist in the presence of constraints are investigated for classes of constraints broad enough to encompass all those encountered in practice. Attention is focussed on principal waves, and results are presented for the growth of the amplitudes of such waves first for fronts of arbitrary curvature, and subsequently by specialisation for plane, cylindrical and spherical waves travelling in material which has undergone one-dimensional plane deformation, cylindrically symmetric and spherically symmetric deformation, respectively.     

On the hysteresis of wire cables in Stockbridge dampers

Stockbridge dampers are used e.g. for reducing wind-excited oscillations due to vortex shedding in conductors of overhead lines. In these dampers, mechanical energy is dissipated in wire cables (“damper cables”). The damping mechanism is due to statical hysteresis resulting from Coulomb (dry) friction between the individual wires of the cable undergoing bending deformation. Systems with statical hysteresis can be modelled by means of Jenkin elements arranged in parallel, consisting of linear springs and Coulomb friction elements. The damper cable is a continuous system and damping takes place throughout the whole length of the cable, so that distributed Jenkin elements are used. The local mechanical properties of the wire cable are identified experimentally in the time domain. In particular, the moment–curvature relation is determined experimentally at every location of the wire cable subjected to dynamic flexural deformations. Using such a model for the damper cables, the equations of motion can be formulated for a Stockbridge damper, and discretization of the damper cable leads to a system of non-linear ordinary differential equations. In order to test this dynamical model of a Stockbridge damper we compute impedance curves and compare them to experimental results.     

VISCOELASTIC CONSTITUTIVE MODEL RELATED TO DEFORMATION OF INSECT WING UNDER LOADING IN FLAPPING MOTION

Flexible insect wings deform passively under the periodic loading during napping flight. The wing flexibility is considered as one of the specific mechanisms on improving insect flight performance. The constitutive relation of the insect wing material plays a key role on the wing deformation, but has not been clearly understood yet. A viscoelastic constitutive relation model was established based on the stress relaxation experiment of a dragonfly wing (in vitro). This model was examined by the finite element analysis of the dynamic deformation response for a model insect wing under the action of the periodical inertial force in flapping. It is revealed that the viscoelastic constitutive relation is rational to characterize the biomaterial property of insect wings in contrast to the elastic one. The amplitude and form of the passive viscoelastic deformation of the wing is evidently dependent on the viscous parameters in the constitutive relation.     

Hydrodynamic and Geometric Stiffening Effects on the Out-of-Plane Waves of Submerged Cables

This study focuses on the relative importance of two sources of nonlinearities affecting submerged cable response. The first of these is the added fluid damping offered by the surrounding medium while the second is the geometric stiffening offered by the cable through finite extensions of its centerline. The contribution of each nonlinear effect, taken separately and in tandem, is evaluated herein through the study of structural waves that form in the (out-of-plane) direction normal to the cable equilibrium plane.Numerical solutions are pursued herein using a finite difference algorithm which is brought to bear on two nonlinear cable/fluid models including: (1)~a nonlinear submerged cable model in which hydrodynamic drag is the sole nonlinear mechanism (referred to herein as the 'nonlinear drag model'); and (2)~a nonlinear submerged cable model in which hydrodynamic drag and geometric stiffening are both active nonlinear mechanisms (the 'nonlinear elastic-drag model'). Numerical solutions for propagating cable waves are developed for the case of a long suspension subjected to a concentrated harmonic excitation source. Conclusions are subsequently drawn regarding the spatial decay of the resulting out-of-plane waves and the dynamic cable tension induced by these waves. The effect of these two nonlinear mechanisms is further explored through the analysis of two additional, linear models: (3)~a simple linear taut string model without drag (the 'simple model'); and (4)~a linear taut string model with linear drag (the 'linear drag model'). The results of all models are critically compared and the range of validity of the linear/cable fluid models are assessed.     

A study of blast wave loading on cantilevers

A study has been made of the response of elastic-plastic and brittle circular-cross-section cantilevers when subjected to blast wave loading. It is demonstrated how the deformation or failure of such cantilevers enables them to be used as blast wave gauges. In addition, the deformation of cantilever-type structures can be used to assess the characteristics of accidental explosions. Two numerical models have been developed to describe the deformation of a dynamically loaded cantilever. Both models assume that the plastic deformation is localized in a region near the fixed end, and that the loading force is a function of the dynamic-pressure time-history and a variable drag coefficient, which depends on the Reynolds number, Mach number and angle of attack of each discretized element of the cantilever. The first model assumes a rigid-plastic response of the cantilever. It was found that this model accurately described the response of cantilevers made of 50/50 lead/tin alloy. It overestimated the deformation of cantilevers made of more elastic materials when exposed to blast waves from high explosives and in a shock tube. The second model assumes an elastic-plastic response. The algorithm is based on the premise that the elastic curvature of the cantilever is limited by the plastic yield stress of the material and that as the curvature approaches this limit the cantilever rotates by the amount needed to keep the curvature constant and equal to this maximum. It has been shown that this algorithm minimizes the curvature of the cantilever at the base. This model provided good predictions of the deformation of cantilevers made of aluminum and steel. The numerical models were evaluated by studying the response of cantilevers exposed to shock waves in a shock tube, and to the blast waves from two explosions of ammonium-nitrate/fuel-oil charges of approximately 2.5 kt. The response to the shock tube flows was recorded by high speed photography which showed good agreement between the observed modes of deflection and those predicted by the elastic-plastic model. The models also provided good predictions of the deformation of a wide range of cantilevers, made of a variety of materials and of different diameters and lengths, when exposed to the free field blast waves. It is demonstrated how the numerical models can be used to determine the type of cantilever that might be used as a gauge for monitoring the blast wave from an explosion, or for evaluating the deformation of a cantilever exposed to the blast wave from an accidental explosion so as to characterize the explosion. Received 23 September 1996 / Accepted 11 November 1996     

考虑阻尼的无限长小垂度缆索弹性波的传播

缆索结构被广泛应用于电气、土木、海洋和航空工程等领域,随着缆索在工程中的应用长度越来越长,高阶振动越来越明显,研究时应该考虑扰动沿着缆索的传播.现有对缆索弹性波传播的研究中,通常不考虑阻尼项,然而阻尼对于波的传播有着重要影响.文章考虑阻尼的影响,发展了包含阻尼项的三维弹性缆索运动方程.通过求解上述含阻尼项的运动方程,分别考察了面内面外弹性波的频率关系、相速度和群速度等自由传播特性,进而通过计算无限长弹性缆索在初始余弦型脉冲作用下的位移响应,分析扰动沿着该缆索的传播规律,考察波的色散现象以及阻尼对于缆索弹性波传播的影响.结果表明,考虑阻尼后,面内波和面外波均为色散波,面内波在曲率的作用下,为高度色散波.此外,在阻尼的影响下,波的峰值在传播过程不断减小,且波的后缘端点响应总是高于前缘端点响应.     

Dynamic stability of spinning viscoelastic cylinders at finite deformation

The study of spinning axisymmetric cylinders undergoing finite deformation is a classic problem in several industrial settings – the tire industry in particular. We present a stability analysis of spinning elastic and viscoelastic cylinders using ARPACK to compute eigenvalues and eigenfunctions of finite element discretizations of the linearized evolution operator. We show that the eigenmodes correspond to N-peak standing or traveling waves for the linearized problem with an additional index describing the number of oscillations in the radial direction. We find a second hierarchy of bifurcations to standing waves where these eigenvalues cross zero, and confirm numerically the existence of finite-amplitude standing waves for the nonlinear problem on one of the new branches. In the viscoelastic case, this analysis permits us to study the validity of two popular models of finite viscoelasticity. We show that a commonly used finite deformation linear convolution model results in non-physical energy growth and finite-time blow-up when the system is perturbed in a linearly unstable direction and followed nonlinearly in time. On the other hand, Sidoroff-style viscoelastic models are seen to be linearly and nonlinearly stable, as is physically required.     

一种适合橡胶类材料的非线性粘弹性本构模型

借助非线性流变模型建立大变形情况非线性粘弹材料的本构关系,考虑到大多数橡胶类材料具有的几乎不可压缩性,以及体积响应和剪切响应的流变性能不同,将变形梯度乘法分解为等容部分和体积变形部分,给出了一种适合橡胶类材料的非线性粘弹性本构模型,并模拟了粘滞效应。对于极快或极慢的过程,该模型退化为橡胶弹性理论;在小变形情况下退化为经典的广义Maxwell粘弹性材料。模型与热力学第二定律相容,适合于大规模数值分析。     

On the large deflections of linear viscoelastic beams

The quasi-static response and the stored and dissipated energies due to large deflections of a slender inextensible beam made of a linear viscoelastic material and subjected to a time-dependent inclined concentrated load at the free end are investigated. The beam cross-section is considered prismatic, the self-weight is disregarded and the material is initially stress free. The set of four first-order non-linear partial integro-differential equations obtained from geometrical compatibility, equilibrium of forces and moments, and linear viscoelastic constitutive relation is numerically solved using a one-parameter shooting method combined with a fourth-order Runge-Kutta algorithm. An analytical expression is derived to divide the energy supplied by the external load into conserved and dissipated parts. For the case study presented, a three-parameter solid linear viscoelastic constitutive model is employed and a step load is applied. The variables are made non-dimensional, so four parameters govern the problem: the ratio between the final and initial relaxation moduli, the load magnitude, the angle of inclination and the unloading time. A finite-element model is also performed to compare and validate the analytical and numerical formulations. Results are presented for encastré curvature and tip displacement versus time, geometrical configuration, load versus tip displacement, total work done by the external force, stored and dissipated energies versus time, energy per unit length along arc length for three times and versus time for two beam sections.     

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.     

Nonlinear seismic waves: A model for site effects

The aim of this paper is to propose a possible mathematical model of site effects that occur when seismic waves propagate through a sediment filled basin. The model is based on the mechanical properties of the medium (that we consider as a granular material) through which the seismic waves propagate. By looking for asymptotic solutions having the features of a progressive wave, we derive an evolution equation which is a modified Korteweg–deVries–Burgers equation containing also a nonlinear dissipative term. This equation is integrated numerically and the modelled site amplification is evaluated by using the smoothed spectral ratio between the propagated profile of the wave and the initial one.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.     

Nonlinear Fractional Order Viscoelasticity at Large Strains

In this paper, we formulate a fractional order viscoelastic model for large deformations and develop an algorithm for the integration of the constitutive response. The model is based on the multiplicative split of the deformation gradient into elastic and viscous parts. Further, the stress response is considered to be composed of a nonequilibrium part and an equilibrium part. The viscous part of the deformation gradient (here regarded as an internal variable) is governed by a nonlinear rate equation of fractional order. To solve the rate equation the finite element method in time is used in combination with Newton iterations. The method can handle nonuniform time meshes and uses sparse quadrature for the calculations of the fractional order integral. Moreover, the proposed model is compared to another large deformation viscoelastic model with a linear rate equation of fractional order. This is done by computing constitutive responses as well as structural dynamic responses of fictitious rubber materials.