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.     

梯度半空间梯度覆层中的Love波

对功能梯度弹性半空间上覆盖一层功能梯度材料中的Love波的频散问题进行了研究,给出 了Love波频散方程的一般形式. 对功能梯度弹性半空间和功能梯度覆层的反平面剪切波的运 动控制方程进行了求解,给出了半空间和覆盖层的位移、应力解析解,给出了Love波在该解析 解下的频散方程. 以覆盖层的剪切弹性模量和质量密度均呈指数函数变化,半空间的剪切弹 性模量和质量密度均呈抛物线变化为例,利用迭代方法对频散方程进行了求解,给出了频散 曲线. 结果显示：在最低阶振型频散曲线中出现截止频率.     

Love waves in functionally graded piezoelectric materials

To investigate the features of Love waves in a layered functionally graded piezoelectric structure, the mathematical model is established on the basis of the elastic wave theory, and the WKB method is applied to solve the coupled electromechanical field differential equation. The solutions of the mechanical displacement and electrical potential function are obtained for the piezoelectric layer and elastic substrate. The dispersion relations of Love waves are deduced for electric open and short cases on the free surface respectively. The actual piezoelectric layer–elastic substrate systems are taken into account, and some corresponding numerical examples are proposed comparatively. Thus, the effects of the gradient variation about material constants on the phase velocity, the group velocity, the coupled electromechanical factor and the cutoff frequency are discussed in detail. So the propagation behaviors of Love waves in inhomogeneous medium is revealed, and the dispersion and the anti-dispersion are analyzed. The conclusions are significant both theoretically and practically for the surface acoustic wave devices.     

Explicit expression of polarization vector for surface waves,slip waves,Stoneley waves and interfacial slip waves in anisotropic elastic materials

We present explicit expression of the polarization vector for surface waves and slip waves in an anisotropic elastic half-space, and Stoneley waves and interfacial slip waves in two dissimilar anisotropic elastic half-spaces. An unexpected result is that, in the case of interfacial slip waves, the polarization vector for the material in the half-space _x2≥0$x_2\ge 0$ does not depend explicitly on the material property in the half-space _x2≤0$x_2\le 0$. It depends on the material property in the half-space _x2≤0$x_2\le 0$ implicitly through the interfacial slip wave speed υ$\upsilon$. The same is true for the polarization vector for the material in the half-space _x2≤0$x_2\le 0$.     

The matrix sign function for solving surface wave problems in homogeneous and laterally periodic elastic half-spaces

The matrix sign function is shown to provide a simple and direct method to derive some fundamental results in the theory of surface waves in anisotropic materials. It is used to establish a shortcut to the basic formulas of the Barnett–Lothe integral formalism and to obtain an explicit solution of the algebraic matrix Riccati equation for the surface impedance. The matrix sign function allows the Barnett–Lothe formalism to be readily generalized for the problem of finding the surface wave speed in a periodically inhomogeneous half-space with material properties that are independent of depth. No partial wave solutions need to be found; the surface wave dispersion equation is formulated instead in terms of blocks of the matrix sign function of i$i$ times the Stroh matrix.     

Propagation of Love waves in non-homogeneous substratum over initially stressed heterogeneous half-space

The paper studies the propagation of Love waves in a non-homogeneous substratum over an initially stressed heterogeneous half-space. The dispersion equation of phase velocity is derived. The velocities of Love waves are calculated numerically as a function of kH and presented in a number of graphs, where k is the wave number, and H is the thickness of the layer. The case of Gibson’s half-space is also considered. It is observed that the speed of Love waves is finite in the vicinity of the surface of the half-space and vanishes as the depth increases for a particular wave number. It is also observed that an increase in compressive initial stresses causes decreases of Love waves velocity for the same frequency, and the tensile initial stress of small magnitude in the half-space causes increase of the velocity.     

A Note on Perturbation Formulae for the Surface-Wave Speed Due To Perturbations in Material Properties

In a recent paper by Tanuma and Man, a two-term asymptotic formula was derived for the speed of surface waves propagating in an anisotropic elastic half-space whose elastic moduli differ only slightly from those for a (base) isotropic elastic material. This formula disagrees with that derived by Delsanto and Clark in an earlier paper using a different method. In this short note, we use a simple procedure to derive another two-term asymptotic formula for the surface-wave speed. Our formula takes the same compact form even if the base material is generally anisotropic. We show that when an error in the work of Delsanto and Clark is corrected, the three different methods do give equivalent results.      

Anisotropic elastic materials capable of a three-dimensional deformation (static or dynamic) with only one displacement component and uncoupling of all three displacement components

It is shown that there are anisotropic elastic materials that are capable of a non-uniform three-dimensional deformation with only one displacement component. For wave propagation, the equation of motion can be cast in the form of the differential equation for acoustic waves. For elastostatics, the equation of equilibrium reduces to Laplace’s equation. The material can be monoclinic, orthotropic, tetragonal, hexagonal or cubic. There are also anisotropic elastic materials that uncouple all three displacement components. The governing equation for each of the uncoupled displacement can be cast in the form of the differential equation for acoustic waves in the case of dynamic or Laplace’s equation in the case of static. The material can be orthotropic, tetragonal, hexagonal or cubic.     

Stroh formulation for a generally constrained and pre-stressed elastic material

The Stroh formalism is essentially a spatial Hamiltonian formulation and has been recognized to be a powerful tool for solving elasticity problems involving generally anisotropic elastic materials for which conventional methods developed for isotropic materials become intractable. In this paper we develop the Stroh/Hamiltonian formulation for a generally constrained and prestressed elastic material. We derive the corresponding integral representation for the surface-impedance tensor and explain how it can be used, together with a matrix Riccati equation, to calculate the surface-wave speed. The proposed algorithm can deal with any form of constraint, pre-stress, and direction of wave propagation. As an illustration, previously known results are reproduced for surface waves in a pre-stressed incompressible elastic material and an unstressed inextensible fibre-reinforced composite, and an additional example is included analyzing the effects of pre-stress upon surface waves in an inextensible material.     

Characteristics of elastic waves in FGM spherical shells,an analytical solution

In this paper, we investigate the propagation characteristics of elastic guided waves in FGM spherical shells with exponentially graded material in the radial direction. A new separation of variables technique to displacements is proposed to convert the governing equations of the wave motion to the second-order ordinary differential equations with variable coefficients. By further a variables transform technique, these equations are transformed to the Whittaker equations so that analytic solutions can be obtained. For the spherical shell case, by satisfying the traction-free boundary conditions on both the inner and outer surfaces of the shell, we obtain the dispersion equations, which show that both the SH and Lamb-type wave modes are generated in the structure. The calculated dispersion curves in the functionally graded shell demonstrate a clear influence of the gradient coefficient as compared to those of the homogeneous shell, with the Lamb-type waves more sensitive to the gradient coefficient. The mode shapes and the distributions of stresses in the shells for various gradient coefficients are also presented to illustrate their dependence on the gradient coefficient.     

Perturbation Formula for Phase Velocity of Rayleigh Waves in Prestressed Anisotropic Media

Herein we consider Rayleigh waves propagating along the free surface of a macroscopically homogeneous, anisotropic, prestressed half-space. We adopt the formulation of linear elasticity with initial stress and assume that the deviation of the prestressed anisotropic medium from a comparative ‘unperturbed’, unstressed and isotropic state, as formally caused by the initial stress and by the anisotropic part of the incremental elasticity tensor, be small. No assumption, however, is made on the material anisotropy of the incremental elasticity tensor. With the help of the Stroh formalism, we derive a first-order perturbation formula for the shift of phase velocity of Rayleigh waves from its comparative isotropic value. Our perturbation formula does not agree totally with that which was derived some years ago by Delsanto and Clark, and we provide another argument as further support for our version of the formula. According to our first-order formula, the anisotropy-induced velocity shifts of Rayleigh waves, taken in totality of all propagation directions on the free surface, carry information only on 13 elastic constants of the anisotropic part of the incremental elasticity tensor. The remaining eight elastic constants are those which would become zero if were monoclinic with the two-fold symmetry axis normal to the free surface of the material half-space in question.     

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 on a Kelvin-Voigt Viscoelastic Half-Space

In this paper we consider the propagation of Rayleigh surface waves in an exponentially graded half-space made of an isotropic Kelvin-Voigt viscoelastic material. Here we take into account the effect of the viscoelastic dissipation energy upon the corresponding wave solutions. As a consequence we introduce the damped in time wave solutions and then we treat the Rayleigh surface wave problem in terms of such solutions. The explicit form of the secular equation is obtained in terms of the wave speed and the viscoelastic inhomogeneous profile. Furthermore, we use numerical methods and computations to solve the secular equation for some special homogeneous materials. The results sustain the idea, existent in literature on the argument, that there is possible to have more than one surface wave for the Rayleigh wave problem.     

梯度介质半空间Rayleigh面波传播性能研究

对剪切弹性模量沿深度以指数函数变化的非均质半空间，本文用摄动法得到了Rayleigh面波的波函数解答及相速度方程。以不同金属与陶瓷复合而成的几种梯度材料为例，用数值方法求解了相速度方程，给出了相应的波的弥散曲线，结果表明，梯度介质半空间自由表面附近的Rayleigh波通常有两种不同的弥散形式，即正常弥散和非正常弥散。     

Effect of rigid boundary on propagation of torsional surface waves in porous elastic layer

The paper presents the effect of a rigid boundary on the propagation of torsional surface waves in a porous elastic layer over a porous elastic half-space using the mechanics of the medium derived by Cowin and Nunziato (Cowin, S. C. and Nunziato, J. W. Linear elastic materials with voids. Journal of Elasticity, 13(2), 125–147 (1983)). The velocity equation is derived, and the results are discussed. It is observed that there may be two torsional surface wave fronts in the medium whereas three wave fronts of torsional surface waves in the absence of the rigid boundary plane given by Dey et al. (Dey, S., Gupta, S., Gupta, A. K., Kar, S. K., and De, P. K. Propagation of torsional surface waves in an elastic layer with void pores over an elastic half-space with void pores. Tamkang Journal of Science and Engineering, 6(4), 241–249 (2003)). The results also reveal that in the porous layer, the Love wave is also available along with the torsional surface waves. It is remarkable that the phase speed of the Love wave in a porous layer with a rigid surface is different from that in a porous layer with a free surface. The torsional waves are observed to be dispersive in nature, and the velocity decreases as the oscillation frequency increases.     

Wave propagation in thermally conducting linear fibre-reinforced composite materials

The propagation of plane waves in a fibre-reinforced, anisotropic, generalized thermoelastic media is discussed. The governing equations in x–y plane are solved to obtain a cubic equation in phase velocity. Three coupled waves, namely quasi-P, quasi-SV and quasi-thermal waves are shown to exist. The propagation of Rayleigh waves in stress free thermally insulated and transversely isotropic fibre-reinforced thermoelastic solid half-space is also investigated. The frequency equation is obtained for these waves. The velocities of the plane waves are shown graphically with the angle of propagation. The numerical results are also compared to those without thermal disturbances and anisotropy parameters.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R$R$-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S$S$-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.     

Plane waves in a hypoplastic half-space under periodic boundary conditions

The paper presents a numerical study of the propagation of plane waves in a half-space occupied by a granular material, with periodic boundary conditions for velocity or stresses prescribed at the boundary of the half-space. The constitutive behaviour of the material is described by a simplified hypoplastic equation which takes into account different values of the stiffness for different directions of deformation, and the coupling between shear and volumetric strains owing to dilatancy. These two features are responsible for a nonlinear character of longitudinal waves and for the generation of longitudinal motion by transverse disturbances. It is shown that longitudinal and transverse boundary disturbances produce qualitatively the same longitudinal waves at large distances from the boundary. As a longitudinal wave propagates, the amplitude of oscillations decreases and eventually vanishes, resulting in a single non-oscillating wave.Received: 10 September 2002, Accepted: 31 March 2003 Correspondence to: Y. A. Berezin     

功能梯度压电夹层结构中SH波的传播

研究了功能梯度压电上、下半空间和均匀压电层组成的夹层结构中SH波的传播性能,上、下功能梯度半空间的材料性能沿垂直于界面方向以指数函数形式变化。首先推导了SH传播时电弹场的解析解,然后利用界面条件得到了行列式形式的频散方程。基于推导的频散方程,通过数值算例表明了材料性能梯度变化、压电层厚度和材料组合方式对相速度的影响,结果对功能梯度压电材料在声波器件中的应用有参考价值。