A note on plane wave propagation in a linear viscoelastic solid

A new formulation of the governing equations for one spatial dimension wave propagation in a linear viscoelastic solid with more than one discrete relaxation time is proposed. The resulting system of three equations is treated as a strictly hyperbolic system of first order hyperbolic partial differential equations and the method of characteristics is adapted to obtain numerical solutions. Results are presented for a solid with two discrete relaxation times and are compared with those obtained from a predictor-corrector scheme.     

Rayleigh wave propagation in a viscoelastic half-space

Summary Rayleigh waves excited by an impulsive force imbedded in a linear viscoelastic half-space are synthesized by applying an approximate inversion of the Fourier transform which yields reliable results. The method is general enough and can be applied to general models of viscoelasticity described by the Boltzmann superposition principle, with a relaxation or creep function given analytically or numerically in the time or frequency domain. Illustrations are given in cases of simple and complicated models of viscoelasticity.     

Plane defect-field wave propagation in a viscoelastic medium

The wave solutions of the system of equations describing the defect field in a viscoelastic medium are studied within the framework of the defect field theory. The refraction and absorption coefficients and the velocities of propagation of the elastic continuum and the defect continuum waves are determined. Features of the correlation between various waves are analyzed.     

Transient wave propagation in a viscoelastic sandwich plate

Summary This paper examines the propagation of coupled P and S waves in a three layer sandwich plate consisting of a viscoelastic core bounded by identical elastic outer layers. The waves are initiated by an impulsive normal line load applied to the upper surface of the plate. The response of the top surface is computed for various values of the creep and relaxation time constants of the core. The results indicate that, for some values of the time constants, the viscoelastic core provides effective damping of the transients, while for others the damping is virtually non-existent. In each case the dominant contribution to the far-field solution comes from the Rayleigh wave.     

Some exact wave propagation solutions in viscoelastic materials

Summary A superposition principle obtained recently by the author, reduces the viscoelastic wave problem to a static elastic problem, an elastic eigenvalue problem and an integrodifferential equation of theVolterra type involving time only. This principle is utilized to obtain exact solutions to a plane, a cylindrical, and a two-dimensional wave problem in a viscoelastic material with constantPoisson's ratio.

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Zusammenfassung Vom Verfasser wurde, kürzlich ein Überlagerungsprinzip hergeleitet, das das viskoelastische Wellenproblem auf ein statisches elastisches Problem, ein elastisches Eigenwertproblem und eineVolterrasche Integrodifferentialgleichung, die nur mehr die Zeit enthält, zurückführt. Dieses Prinzip wird nunmehr benützt, um exakte Lösungen eines ebenen, eines zylindersymmetrischen und eines zweidimensionalen Wellenproblems in einem viskoelastischen Material mit konstanterPoissonzahl zu erhalten.

     

Axial wave propagation in viscoelastic bars using a new finite-element-based method

A new solution for the finite-element implementation of the axial vibration of viscoelastic bars is proposed. The effect of viscoelasticity on the dynamic response is accounted for by a convolution integral, avoiding the difficulties associated with the high-order time derivatives used in conventional models for viscoelasticity. A convenient but excellent numerical approximation for the obtained convolution integral is proposed. This numerical approximation allows easy implementation of the finite-element procedure in the time domain as it only introduces additional terms to the mass matrix and the force vector. The additional terms are calculated from quantities obtained in the previous time step. To validate the proposed numerical procedure, a few simple examples are presented and solved by the existing direct methods as well as the new alternative method. The examples include the computation of the dynamic axial response of a viscoelastic bar fixed at one end and subject to a step or sinusoidally varying load at the other end. It is concluded that the new method is valid and works satisfactorily.     

Love wave propagation in a solid with a cold-worked surface layer

In this paper, experiments show Love wave generation along the top surface of used railroad rail, where the shear wave velocity has been slightly reduced by the cold-working of wheel passage for years. The rf pulses used in the experiments have about 10 µs duration and a relatively narrow frequency spectrum. The group velocity of the Love wave is found to have a strong dependence on the carrier frequency over the tested range of 0.45–3.1 MHz. Application of the seismological one-layered model to the experimental measurements yields an NDE technique for the elastic properties and the thickness of the cold-worked surface layer. The results are interpreted on the basis of a destructive observation by micro-Vickers hardness testing.     

Nonlinear axial shear wave propagation in a hyperelastic incompressible solid

Summary This paper is concerned with the propagation of axial shear waves in an incompressible isotropic hyperelastic solid, whose strain energy function is expressible as a power series in (I ₁-3) and (I ₂-3) whereI ₁ andI ₂ are the first and second basic invariants of the left Cauchy-Green tensorB. Numerical solutions are presented for problems of wave propagation produced by a step function application, or a finite duration pulse, of axial shear stress at the surface of a cylindrical cavity in an unbounded medium. A modification of MacCormack's finite difference scheme [1] is proposed and is used to obtain these solutions along with a procedure for the determination of the position of the shock front for the step function application.An estimate of the breaking time of a wave, obtained from a procedure proposed by Whitham [2], is compared with the numerical results. The dissipation of mechanical energy due to shock propagation is considered.With 4 Figures     

Rayleigh wave propagation in a solid with a cold-worked surface layer

The propagation of the Rayleigh surface wave is experimentally studied along the top surface of used railroad rail under conditions where ultrasonic pulses have carrier frequencies ranging from 0.4 to 3.0 MHz and approximately 10 µs duration. The generation of the first higher (M ₂₁ or Sezawa) mode as well as the fundamental (M ₁₁) mode and their dispersion properties are observed. These phenomena are attributable to the presence of the cold-worked surface layer caused by the wheel passage. It is shown that a theoretical model of a single layer overlying a half space, whose elastic constants are determined by a destructive method, yields results which agree with the dispersion curves obtained experimentally. On the basis of this one-layered model, an inversion method to estimate the layer thickness and its elastic constants is discussed.     

Guided wave propagation in an elastic hollow cylinder coated with a viscoelastic material

The propagation of ultrasonic guided waves in an elastic hollow cylinder with a viscoelastic coating is studied. The principle motivation is to provide tools for performing a guided wave, nondestructive inspection of piping and tubing with viscoelastic coatings. The theoretical boundary value problem is solved that describes the guided wave propagation in these structures for the purpose of finding the guided wave modes that propagate with little or no attenuation. The model uses the global matrix technique to generate the dispersion equation for the longitudinal modes of a system of an arbitrary number of perfectly bonded hollow cylinders with traction-free outer surfaces. A numerical solution of the dispersion equation produces the phase velocity and attenuation dispersion curves that describe the nature of the guided wave propagation. The attenuation dispersion curves show some guided wave modes that propagate with little or no attenuation in the coated structures of interest. The wave structure is examined for two of the modes to verify that the boundary conditions are satisfied and to explain their attenuation behavior. Experimental results are produced using an array of transducers positioned circumferentially around the pipe to evaluate the accuracy of the numerical solution.     

Shear wave propagation in vertically heterogeneous viscoelastic layer over a micropolar elastic half-space

This article deals with the propagation of SH-wave in a vertically heterogeneous viscoelastic layer lying over a micropolar elastic half-space. Dispersion and damping equations are obtained analytically in closed form. Phase and damped velocities are computed numerically and depicted by means of a graph to exhibit the substantial effect of heterogeneity, viscoelasticity (internal friction), and micropolar parameter. As a special case of the problem, it is found that deduced dispersion relation is well in agreement to the classical-Love wave equation and damping equation vanishes identically for the isotropic case. Influence of micropolarity present in the medium of half-space is highlighted through comparative study.     

Dynamic-contact stiffness at the interface between a vibrating rigid sphere and a semi-infinite viscoelastic solid

Vibration of a rigid sphere contacting a semiinfinite viscoelastic solid is studied theoretically. The rigid sphere is subjected to an oscillating force while being compressed in the vertical direction by a static force. The contact-pressure distribution and contact radius at the interface vary with the oscillation. Assuming sufficiently small oscillating force, we derive the dynamic-contact-pressure distribution with the constant contact radius, which establishes dynamic contact stiffness between the sphere and the viscoelastic solid. Numerical calculations show the influence of vibration frequency, contact radius, Poisson's ratio, and the damping characteristic of the solid.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

Finite amplitude spherically symmetric wave propagation in a compressible hyperelastic solid

Summary Some aspects of the wave propagation, resulting from the spherically symmetric expansion of a thick walled hyperelastic shell and the limiting case of expansion of a cavity in an unbounded medium, are investigated. It is assumed that the shell is isotropic and uniform in the natural reference state and its strain energy function is a particular compressible generalization of that for the neo-Hookean solid. The response of a compressible shell, due to a spatially uniform time dependent application of internal pressure, is compared with that for the neo-Hookean shell taken as a limiting case of the compressible shell. This is also done for an unbounded medium.A finite difference method which uses the relation along one of the families of characteristics is used to obtain numerical results. In order to implement this method the governing equations are expressed as a system of first order partial differential equations in conservation form.With 9 Figures     

Wave propagation in linear viscoelastic composites modelled as interpenetrating solid continua

This work concerns the behavior of shock waves and higher order discontinuities which propagate in a viscoelastic composite which is modelled as a mixture ot two interpenetrating solid continua. The response of the composite is assumed to be linear. The propagation conditions for shock and higher order waves are derived and it is shown that, in general, six waves of any order may propagate in any direction. It is shown that the amplitudes of all waves decay as they propagate. Detailed results are presented for isotropic composites.     

在含气孔粘弹媒质中非线性声传播特性

1引言: 共振型消声是实现水下吸声的一种基本形式.结构共振消声是通过在基底粘弹(通常为橡胶类粘弹材料)材料中嵌入声学结构--各种类型和大小的空腔,利用空腔的共振吸收增加损耗.本文运用等效媒质法,着重对含气孔粘弹材料的二阶非线性声场做了研究.     

狭缝空腔间粘弹性体中弹性波传播特性的研究

利用笛卡尔坐标下的Kelvin-Voigt线性粘弹性模型,研究了无限长粘弹性狭缝通道中波的传播和衰减。利用自适应绕数求根方法,针对具有狭缝形空腔周期性分布覆盖层的吸声问题,求得频散方程的根,并且得到无限长狭缝空腔中相速度频散曲线和衰减曲线。分析了各阶模式对无限长粘弹性狭缝结构中波传播的相速度和衰减的影响。通过引入复粘弹性模量,粘弹性波动方程具有弹性波动方程相对应的形式,计算中可用复值粘弹性模量代替相应的弹性模量。粘弹性狭缝结构中所有波传播模式均存在衰减,高阶波在某个频率以下衰减非常大,而在该频率以上逐渐减小;最低阶模式的相速度与材料无损耗的情况非常接近。粘弹性狭缝结构中波传播的衰减特性不仅与狭缝的结构参数有关,还与粘弹性材料本身的复弹性模量有关。     

Studying nonlinear thermomechanical wave propagation in a viscoelastic layer based upon the Lord-Shulman theory

Abstract

Based on the Lord–Shulman (L-S) theory, the thermally nonlinear coupled thermo-viscoelasticity of a layer is analyzed using a numerical approach. Using the Kelvin-Voigt theory of viscoelasticity in conjunction with the L-S theory, the coupled governing equations are first derived in variational form. Then, the variational differential quadrature (VDQ) method is applied to solve the problem numerically. Parametric studies are presented in order to reveal the effects of viscoelastic parameter, intensity of heat flux and relaxation time on the propagation of displacement, temperature and stress waves. Furthermore, the influence of considering nonlinearity in the energy equation is analyzed.     

Rayleigh-type wave propagation on a transversely isotropic viscoelastic layer with yielding and rigid foundations

The present study investigates the propagation of Rayleigh-type wave in a transversely isotropic viscoelastic layer in effect of yielding foundation and rigid foundation in two different cases for two considered models. Numerical computation and graphical demonstration have been carried out for the case when layer is comprised of transversely isotropic viscoelastic Mesaverde clay shale material (Model I) and simply isotropic viscoelastic material (Model II). Closed-form expression of phase velocity and damped velocity for both the cases are deduced analytically. Obtained result is found in well agreement to the established standard results existing in the literature. Significant effect of dilatational viscoelasticity, volume viscoelasticity and yielding parameter on phase and damped velocities for both the considered models has been traced out. The comparative study has been performed to unravel the effect of viscoelasticity over elasticity and anisotropy over isotropy in the context of the present problem. Moreover, comparison of phase and damped velocities for the case of layer with stress-free foundation, layer with rigid foundation and layer with yielding foundation serve as a major highlight of the present work.     

Determining the thermal stresses in a hollow viscoelastic sphere

For a viscoelastic sphere with spherical cavity, solution of the quasistatic problem of the stresses produced by a nonstationary temperature field reduces to solution of an integral-differential equation whose right side depends on an unknown function of the time. A numerical solution method is described.Translated from Inzhenerno-Fizicheskii Zhurnal, Vol. 17, No. 2, pp. 300–305, August, 1969.