Thermoelastic surface waves on an exponentially graded half-space

In this paper, we consider the propagation of Rayleigh surface waves in a functionally graded isotropic thermoelastic half-space, in which all thermoelastic characteristic parameters exponentially change along the depth direction. The propagation condition is established in the form of a bicubic equation whose coefficients are complex numbers while the analytical solutions (eigensolutions) of the thermoelastodynamic system are explicitly obtained in terms of the characteristic solutions. The concerned solution of the Rayleigh surface wave problem is subsequently expressed as a linear combination of the three eigensolutions while the secular equation is established in an implicit form. The explicit secular equation is written when an isotropic and homogeneous thermoelastic half-space is considered and some numerical simulations are given for a specific material.     

Rayleigh surface waves in the theory of thermoelastic materials with voids

In this paper we analyze the behavior of plane harmonic waves and Rayleigh waves in a linear thermoelastic material with voids. We take into account the damped effects of the thermal field upon the propagation waves. Consequently, the propagation condition is established in the form of an algebraic equation of 9th degree whose coefficients are complex numbers while the eigensolutions of the thermoelastodynamic with voids system are explicitly obtained in terms of the characteristic solutions. We show that the transverse waves are undamped in time and they are not influenced by the thermal and porous effects while the longitudinal waves are all damped in time and they are coupled with the thermal and porous effects. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the eigensolutions in concern. The secular equation is established in an implicit form and afterwards an explicit form is written for an isotropic and homogeneous thermoelastic with voids half-space. Furthermore, we use the numerical methods and computations to solve the secular equation for a specific material.     

Rayleigh waves in orthotropic fluid-saturated porous media

In this paper, we are interested in the propagation of Rayleigh waves in orthotropic fluid-saturated porous media. This problem was investigated by Liu and Liu (2004). The authors have derived the secular equation of the wave but that secular equation is still in implicit form. The main aim of this paper is to derive explicit secular equation of the wave. By employing the method of polarization vector, the secular equations of Rayleigh waves in explicit form is obtained. This equation recovers the dispersion equation of Rayleigh waves propagating in pure orthotropic elastic half-spaces. Remarkably, the secular equation obtained is not a complex equation as the one derived by Liu and Liu, it is a really real equation.     

Rayleigh waves in an isotropic elastic half-space coated by a thin isotropic elastic layer with smooth contact

In the present paper, we are interested in the propagation of Rayleigh waves in an isotropic elastic half-space coated with a thin isotropic elastic layer. The contact between the layer and the half space is assumed to be smooth. The main purpose of the paper is to establish an approximate secular equation of the wave. By using the effective boundary condition method, an approximate, yet highly accurate secular equation of fourth-order in terms of the dimensionless thickness of the layer is derived. From the secular equation obtained, an approximate formula of third-order for the velocity of Rayleigh waves is established. The approximate secular equation and the formula for the velocity obtained in this paper are potentially useful in many practical applications.     

Rayleigh Waves on an Exponentially Graded Poroelastic Half Space

In this paper we consider the propagation of seismic waves in isotropic poroelastic half spaces with continuously varying elastic properties, namely with an exponentially decaying depth profile. The present paper shows that the problem leads naturally to a bicubic equation. We obtain explicit inhomogeneous plane wave solutions in an exponential evanescent form with respect to the depth of half space. Further, these solutions are used to solve the boundary value problem of a Rayleigh surface wave and the secular equation is established. The results obtained theoretically are exemplified for numerical data and represented graphically for a representative poroelastic material.     

无黏流体与固体界面的漏瑞利波

漏瑞利波存在于半无限无黏性流体和半无限固体媒质的界面处. 首先推导流固无限各向同性介质界面处漏瑞利波的特征方程和位移及应力的解析计算公式. 然后结合典型结构通过数值计算研究了漏瑞利波特性以及位移和应力在流体和固体中的分布规律. 数值计算结果表明漏瑞利波的相速度和衰减随流固密度比的增大而增大, 在流固界面上法向位移连续而切向位移不连续. 流固密度比对固体媒质中沿垂直于漏瑞利波的传播方向的位移、正应力和剪应力有比较大的影响,而对沿漏瑞利波的传播方向的正应力几乎没影响. 为利用漏瑞利波的无损检测与评价提供了理论基础.      

Rayleigh waves in an incompressible elastic half-space overlaid with a water layer under the effect of gravity

This paper is concerned with the propagation of Rayleigh waves in an incompressible isotropic elastic half-space overlaid with a layer of non-viscous incompressible water under the effect of gravity. The authors have derived the exact secular equation of the wave which did not appear in the literature. Based on it the existence of Rayleigh waves is considered. It is shown that a Rayleigh wave can be possible or not, and when a Rayleigh wave exists it is not necessary unique. From the exact secular equation the authors arrive immediately at the first-order approximate secular equation derived by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. When the layer is assumed to be thin, a fourth-order approximate secular equation is derived and of which the first-order approximate secular equation obtained by Bromwich is a special case. Some approximate formulas for the velocity of Rayleigh waves are established. In particular, when the layer being thin and the effect of gravity being small, a second-order approximate formula for the velocity is created which recovers the first-order approximate formula obtained by Bromwich [Proc. Lond. Math. Soc. 30:98–120, 1898]. For the case of thin layer, a second-order approximate formula for the velocity is provided and an approximation, called global approximation, for it is derived by using the best approximate second-order polynomials of the third- and fourth-powers.     

Rayleigh lamb waves in micropolar isotropic elastic plate

The propagation of waves in a homogeneous isotropic micropolar elastic cylindrical plate subjected to stress free conditions is investigated. The secular equations for symmetric and skew symmetric wave mode propagation are derived. At short wave limit, the secular equations for symmetric and skew symmetric waves in a stress free circular plate reduces to Rayleigh surface wave frequency equation. Thin plate results are also obtained. The amplitudes of displacements and microrotation components are obtained and depicted graphically. Some special cases are also deduced from the present investigations. The secular equations for symmetric and skew symmetric modes are also presented graphically.     

Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.     

Modeling nonlinear Rayleigh wave fields generated by angle beam wedge transducers—A theoretical study

Nonlinear Rayleigh wave fields generated by an angle beam wedge transducer are modeled in this study. The calculated area sound sources underneath the wedge are used to model the fundamental Rayleigh sound fields on the specimen surface, which are more accurate than the previously used line sources with uniform or Gaussian amplitude distributions. A general two-dimensional nonlinear Rayleigh wave equation without parabolic approximation is introduced and the solutions are obtained using the quasilinear theory. The second harmonic Rayleigh wave due to material nonlinearity is given in an integral expression with these fundamental Rayleigh waves radiated by the wedge transmitter acting as a forcing function. Multi-Gaussian beam (MGB) models are employed to simplify these integral solutions and to extract the diffraction and attenuation correction terms explicitly. The effect of nonlinearity of generating sources on the second harmonic Rayleigh wave fields is taken into consideration; simulation results show that it will affect the magnitude and diffraction correction of the second harmonic waves in the region close to the Rayleigh wave sound sources. This research provides a theoretical improvement to alleviate the experimental restriction on analyzing the effects of diffraction, attenuation and source nonlinearity when using angle beam wedge transducers as transmitters.     

Spherical bubble collapse in viscoelastic fluids

The collapse of a spherical bubble in an infinite expanse of viscoelastic fluid is considered. For a range of viscoelastic models, the problem is formulated in terms of a generalized Bernoulli equation for a velocity potential, under the assumptions of incompressibility and irrotationality. The boundary element method is used to determine the velocity potential and viscoelastic effects are incorporated into the model through the normal stress balance across the surface of the bubble. In the case of the Maxwell constitutive equation, the model predicts phenomena such as the damped oscillation of the bubble radius in time, the almost elastic oscillations in the large Deborah number limit and the rebound limit at large values of the Deborah number. A rebound condition in terms of ReDe is derived theoretically for the Maxwell model by solving the Rayleigh–Plesset equation. A range of other viscoelastic models such as the Jeffreys model, the Rouse model and the Doi-Edwards model are amenable to solution using the same technique. Increasing the solvent viscosity in the Jeffreys model is shown to lead to increasingly damped oscillations of the bubble radius.     

On a 3D moving load problem for an elastic half space

The paper deals with 3D dynamic response of an elastic half-space loaded by a point force moving at a constant speed along a straight line on the surface. The problem is formulated within the framework of the asymptotic hyperbolic–elliptic model developed earlier by two of the authors. The validity of the model is restricted to the range of speeds close to the Rayleigh wave speed. Steady-state near-field solutions are derived in terms of elementary functions. Transient analysis of surface motion illustrates peculiarities of the resonance associated with the Rayleigh wave.     

Plane strain dynamics of elastic solids with intrinsic boundary elasticity, with application to surface wave propagation

In this paper, in a development of the static theory derived by Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853), we establish the equations of motion for a non-linearly elastic body in plane strain with an elastic surface coating on part or all of its boundary. The equations of (linearized) incremental motions superposed on a finite static deformation are then obtained and applied to the problem of (time-harmonic) surface wave propagation on a pre-stressed incompressible isotropic elastic half-space with a thin coating on its plane boundary. The secular equation for (dispersive) wave speeds is then obtained in respect of a general form of incompressible isotropic elastic strain-energy function for the bulk material and a general energy function for the coating material. Specialization of the form of strain-energy function enables the secular equation to be cast as a quartic equation and we therefore focus on this for illustrative purposes. An explicit form for the secular equation is thereby obtained. This involves a number of material parameters, including residual stress and moment in the properties of the coating. It is shown how this equation relates to previous work on waves in a half-space with an overlying thin layer set in the classical theory of isotropic elasticity and, in particular, the significant effect of omission of the rotatory inertia term, even at small wave numbers, is emphasized. Corresponding results for a membrane-type coating, for which the bending moment, inertia and residual moment terms are absent, are also obtained. Asymptotic formulas for the wave speed at large wave number (high frequency) are derived and it is shown how these results influence the character of the wave speed throughout the range of wave number values. A bifurcation criterion is obtained from the secular equation by setting the wave speed to zero, thereby generalizing the bifurcation results of Steigmann and Ogden (Proc. Roy. Soc. London A 453 (1997) 853) to the situation in which residual stress and moment are present in the coating. Numerical results which show the dependence of the wave speed on the various material parameters and the finite deformation are then described graphically. In particular, features which differ from those arising in the classical theory are highlighted.     

Secular Equations for Rayleigh and Stoneley Waves in?Exponentially Graded Elastic Materials of General Anisotropy under the Influence of Gravity

The Stroh formalism is employed to study Rayleigh and Stoneley waves in exponentially graded elastic materials of general anisotropy under the influence of gravity. The 6×6 fundamental matrix N is no longer real. Nevertheless the coefficients of the sextic equation for the Stroh eigenvalue p are real. The orthogonality and closure relations are derived. Also derived are three Barnett-Lothe tensors. They are not necessarily real. Secular equations for Rayleigh and Stoneley wave speeds are presented. Explicit secular equations are obtained when the materials are orthotropic. In the literature, the secular equations for Stoneley waves in orthotropic materials are obtained without using the Stroh formalism. As a result, it requires computation of a 4×4 determinant. The secular equation presented here requires computation of a 2×2 determinant, and hence is fully explicit. A Rayleigh or Stoneley wave exists in the exponentially graded material under the influence of gravity if the wave can propagate in the homogeneous material without the influence of gravity. As the wave number k????, the Rayleigh or Stoneley wave speed approaches the speed for the homogeneous material.     

Rayleigh waves with impedance boundary conditions in anisotropic solids

The paper is concerned with the propagation of Rayleigh waves in an elastic half-space with impedance boundary conditions. The half-space is assumed to be orthotropic and monoclinic with the symmetry plane _x3=0$x_3=0$. The main aim of the paper is to derive explicit secular equations of the wave. For the orthotropic case, the secular equation is obtained by employing the traditional approach. It is an irrational equation. From this equation, a new version of the secular equation for isotropic materials is derived. For the monoclinic case, the method of polarization vector is used for deriving the secular equation and it is an algebraic equation of eighth-order. When the impedance parameters vanish, this equation coincides with the secular equation of Rayleigh waves with traction-free boundary conditions.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Free vibration of microstretch thermoelastic plate with one relaxation time

The propagation of waves in microstretch thermoelastic homogeneous isotropic plate subjected to stress free thermally insulated and isothermal conditions is investigated in the context of conventional coupled thermoelasticity (CT) and Lord and Shulman (L–S) theories of thermoelasticity. The secular equations for both symmetric and skew-symmetric wave mode propagation have been obtained. At short wavelength limits, the secular equations for symmetric and skew-symmetric modes reduce to Rayleigh surface wave frequency equation. The amplitudes of dilatation, microrotation, microstretch and temperature distribution for the symmetric and skew symmetric wave modes are computed analytically and presented graphically for different theories of thermoelasticity. The theoretical and numerical computations are found to be in close agreement.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Steady-state response of axially moving viscoelastic beams with pulsating speed: comparison of two nonlinear models

Principal parametric resonance in transverse vibration is investigated for viscoelastic beams moving with axial pulsating speed. A nonlinear partial-differential equation governing the transverse vibration is derived from the dynamical, constitutive, and geometrical relations. Under certain assumption, the partial-differential reduces to an integro-partial-differential equation for transverse vibration of axially accelerating viscoelastic nonlinear beams. The method of multiple scales is applied to two equations to calculate the steady-state response. Closed form solutions for the amplitude of the vibration are derived from the solvability condition of eliminating secular terms. The stability of straight equilibrium and nontrivial steady-state response are analyzed by use of the Lyapunov linearized stability theory. Numerical examples are presented to highlight the effects of speed pulsation, viscoelascity, and nonlinearity and to compare results obtained from two equations.     

Complex surface rays associated with inhomogeneous skimming and Rayleigh waves

The Rayleigh wave, that propagates at the free surface of semi-infinite anisotropic medium, is composed of three inhomogeneous partial waves, each propagating along the surface with a different attenuation along the depth. Since this wave does not exhibit an attenuation on the surface, let us call it the homogeneous Rayleigh wave. The associated slowness corresponds to the real solution of the Rayleigh dispersion equation. Besides this classical solution, an infinite number of complex solutions of the Rayleigh dispersion equation exits. For such particular Rayleigh waves, the slowness vector, i.e. the identical component on the surface of the slowness of each partial waves, is taken to be complex. Thus, these Rayleigh waves are attenuated on the surface and as shown here, their attenuation is normal to the ray direction (or the energy velocity direction). Similarly to the infinite inhomogeneous plane waves which can be associated with complex rays, we call these waves, inhomogeneous Rayleigh waves. We use the inhomogeneous skimming waves, which are inhomogeneous plane waves, and the inhomogeneous Rayleigh waves to explain differently the usual diffraction phenomena on the free surface which cannot be explained by the real ray theory. For example, the arrival time of the wave packet observed beyond the cusp is in perfect accordance with the arrival time of some specific inhomogeneous Rayleigh waves. We show that these results are in agreement with the computation of the Green function. They apply to the theory of surface waves in linear elastodynamics with intrinsic anisotropy as well as to the theory of surface waves in linearised (incremental) elastodynamics with strain-induced anisotropy (also known as small-amplitude waves superimposed on the large static homogeneous deformation of a non-linear solid).