Finite-amplitude Love waves in a pre-stressed compressible elastic half-space with a double surface layer

The dispersive behavior of finite-amplitude time-harmonic Love waves propagating in a pre-stressed compressible elastic half-space overlaid with two compressible elastic surface layers of finite thickness is investigated. The half-space and layers are made of different pre-stressed compressible neo-Hookean materials. The dispersion relation which relates wave speed and wavenumber is obtained in explicit form. Results for the energy density and energy flux of the waves are also presented. The special case where the interfaces between the layers and the half-space are principal planes of the left Cauchy–Green deformation tensor is also investigated. Numerical results are presented showing the variation of the Love wave speed with the pre-stress and the propagation angle.     

Surface waves and surface stability for a pre-stretched, unconstrained, non-linearly elastic half-space

An unconstrained, non-linearly elastic, semi-infinite solid is maintained in a state of large static plane strain. A power-law relation between the pre-stretches is assumed and it is shown that this assumption is well motivated physically and is likely to describe the state of pre-stretch for a wide class of materials. A general class of strain-energy functions consistent with this assumption is derived. For this class of materials, the secular equation for incremental surface waves and the bifurcation condition for surface instability are shown to reduce to an equation involving only ordinary derivatives of the strain-energy equation. A compressible neo-Hookean material is considered as an example and it is found that finite compressibility has little quantitative effect on the speed of a surface wave and on the critical ratio of compression for surface instability.     

Anti-plane waves near an interface between two piezoelectric half-spaces

We show the existence of certain new waves that can propagate near an interface between two half-spaces of different piezoelectric ceramics, where the interface is modeled by a membrane with the surface/interface elasticity [1]. The current configuration can be reduced into a number of well-known results as special cases, such as Love wave, Bleustein and Gulyaev wave. Together with our previous work for the imperfect interface [2], a full range of consideration of the interface affecting the anti-plane waves is now completed.     

Shear horizontal waves in transversely inhomogeneous plates

Shear horizontal (SH) waves in free and clamped monoclinic plates with an arbitrary inhomogeneity across the plate are considered. Firstly, some general properties of dispersion spectra of, specifically, the SH waves are pointed out. Secondly, analytical and numerical modeling of the SH dispersion branches is presented. Closed-form estimations are compared with exact curves computed for a free plate with continuously varying properties.     

On formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two elastic half-spaces

Formulas for the velocity of Stoneley waves propagating along the loosely bonded interface of two isotropic elastic half-spaces are derived using the complex function method. The derivation also shows that if a Stoneley wave exists, then it is unique. By using the obtained formulas, we can easily reproduce the numerical results previously obtained by Murty [G. S. Murty, Phys. Earth Planet. Interiors 11 (1975), 65–79.] by directly solving the secular equation.     

Surface waves at the interface between an inviscid fluid and a dipolar gradient solid

This paper is about the dispersion analysis of surface waves propagating at the interface between an inviscid fluid and a higher gradient homogeneous elastic solid modelled as a dipolar gradient continuum. In order to compare the results, a second gradient model is also evaluated. The analysis is carried out by finding the roots of the secular equation, and by carefully studying their physical meaning. As it is well known, higher gradient continua are dispersive, i.e. phase and group velocities are frequency dependent. As a consequence, the existence of surface waves will indeed depend on frequency. In order to investigate the behaviour of surface waves in this specific fluid–solid configuration, a complete dispersion analysis is performed, with a particular focus on the frequency range in which the phase velocity of shear waves is lower than the speed of waves of the fluid. Surface waves of the type Leaky Rayleigh and Scholte–Stoneley are observed in this frequency range. This work extends the knowledge on surface waves in the case of higher gradient solids and applications of these results can be found in the field of non-destructive damage evaluation in micro structured materials, composites, metamaterials and biological tissues.     

Piezoelectromagnetic waves in a ceramic plate between two ceramic half-spaces

We analyze the propagation of piezoelectromagnetic waves guided by a plate of polarized ceramics between two ceramic half-spaces. An exact dispersion relation is obtained, which reduces to a few known elastic, electromagnetic, and quasistatic piezoelectric wave solutions in the literature as special cases. Numerical solutions to the equation that determines the dispersion relation show the existence of guided waves. The results are useful for acoustic wave and microwave devices.     

Dispersion effects of extensional waves in pre-stressed imperfectly bonded incompressible elastic layered composites

The effect of an imperfect interface, on time-harmonic extensional wave propagation in a pre-stressed symmetric layered composite is considered. The bimaterial composite consists of incompressible isotropic elastic materials. The shear spring type resistance model employed to simulate the imperfect interface can accommodate the extreme cases of perfect bonding and a fully slipping interface. The dispersion relation obtained by formulating the incremental boundary-value problem and the use of the propagator matrix technique, is analyzed at the low and high wavenumber limits. For the perfectly bonded and imperfect interface cases in the low wavenumber region, only the fundamental mode has a finite phase speed, while other higher modes have an infinite phase speed when the dimensionless wavenumber approaches zero. However, for the fully slipping interface in the low wavenumber region, both the fundamental mode and the next lowest mode have finite phase speeds. In the high wavenumber region, when the dimensionless wavenumber tends to infinity, the phase speeds of the fundamental mode and the higher modes depend on the phase speeds of the surface and interfacial waves and on the limiting phase speed of the composite. An expression to determine the cut-off frequencies is obtained from the dispersion relation. Numerical examples of dispersion curves are presented, where when the material has to be prescribed either Mooney–Rivlin material or Varga material is assumed. The effect of the imperfect interface is clearly evident in the numerical results.     

Analogy between antiplane shear waves in anisotropic and isotropic media

A linear transformation is presented that transforms the fundamental equations of antiplane shear waves in an anisotropic medium into those of an isotropic medium. By this transformation, the solution of the former problem may be easily obtained if the corresponding latter problem has been solved.     

Reflection and transmission of elastic waves at five types of possible interfaces between two dipolar gradient elastic half-spaces

Reflection and transmission of an incident plane wave at five types of possible interfaces between two dipo-lar gradient elastic solids are studied in this paper. First, the explicit expressions of monopolar tractions and dipolar trac-tions are derived from the postulated function of strain energy density. Then, the displacements, the normal derivative of displacements, monopolar tractions, and dipolar tractions are used to create the nontraditional interface conditions. There are five types of possible interfaces based on all possible combinations of the displacements and the normal derivative of displacements. These interfacial conditions with consid-eration of microstructure effects are used to determine the amplitude ratio of the reflection and transmission waves with respect to the incident wave. Further, the energy ratios of the reflection and transmission waves with respect to the incident wave are calculated. Some numerical results of the reflection and transmission coefficients are given in terms of energy flux ratio for five types of possible interfaces. The influences of the five types of possible interfaces on the energy parti-tion between the refection waves and the transmission waves are discussed, and the concept of double channels of energy transfer is first proposed to explain the different influences of five types of interfaces.     

Application of the Ritz–Rayleigh method for Lamb waves in extremely anisotropic media

We propose a numerical approach for the calculation of frequency–dispersion curves in a flat anisotropic waveguide based on the Ritz–Rayleigh method, offering several significant benefits over commonly used analytical and numerical models. The problem is linearized using a tailored functional basis based on Legendre polynomials, utilizing all symmetries provided by the problem to reduce the computational demands, which brings significant benefits for an inverse procedure, and thus for material characterization. The approach is completely general in terms of elastic properties and direction of propagation. As an exemplary calculation, frequency–dispersion curves of phase and group velocity are calculated for a (001)-oriented free-standing film of the Ni–Mn–Ga alloy exhibiting a strong cubic anisotropy. In the case of propagation along the [100] direction, the dispersion curves oscillate strongly, creating pairs of symmetric and antisymmetric modes. Taking advantage of the generality of the approach, the curves are also presented for the case of extreme anisotropy. Sets of curves for directions [110] and [1 0.9 0] (i.e., 3${}^{\circ }$ off [110]), where the pseudo-surface wave exists, are also calculated. High-frequency asymptotes are shown to correspond with surface and bulk modes propagating along the plate.     

PMN-PT夹层弹性半空间结构中SH波的传播Propagation of SH waves in sandwich structures consisting of PMN-PT single crystal layer and elastic half-spaces

分析了弹性上下半空间和PMN‐PT单晶层组成的夹层结构中SH波的传播性质,PMN‐PT单晶沿[011]c方向极化,宏观上呈mm2对称,且晶体沿角度θ方向切割。基于正交各向异性压电材料和各向同性弹性材料的基本方程,得到了夹层结构中SH波传播时行列式形式的频散方程。通过对数值算例进行分析可以看出,PMN‐PT单晶的切割角度和弹性材料属性对结构中的相速度有很大影响,因此波的某些传播性能可以通过材料的设计以及晶体切割的方向来实现,这些结论为声表面波器件的开发和应用提供了理论依据。     

Padé approximations for low-frequency motion in a pre-stressed compressible elastic layer

In this paper, we derive an asymptotic approximation of the low-frequency motion in a pre-stressed compressible elastic layer, valid in both long- and short-wave regimes. For this purpose, we shall use Padé approximations. Comparisons with numerical solutions are provided and show good agreement for all wavelength. Results appropriate to the classical linear isotropic case are also briefly provided.     

A three-layer structure model for the effect of a soft middle layer on Love waves propagating in layered piezoelectric systems

A three-layer structure model is proposed for investigating the effect of a soft elastic middle layer on the propagation behavior of Love waves in piezoelectric layered systems, with "soft" implying that the bulk-shear-wave velocity of the middle layer is smaller than that of the upper sensitive layer. Dispersion equations are obtained for unelectroded and traction-free upper surfaces which, in the limit, can be reduced to those for classical Love waves. Systematic parametric studies are subsequently carried out to quantify the effects of the soft middle layer upon Love wave propagation, including its thickness, mass density, dielectric constant and elastic coefficient. It is demonstrated that whilst the thickness and elastic coefficient of the middle layer affect significantly Love wave propagation, its mass density and dielectric constant have negligible influence. On condition that both the thickness and elastic coefficient of the middle layer are vanishingly small so that it degenerates into an imperfectly bonded interface, the three-layer model is also employed to investigate the influence of imperfect interfaces on Love waves propagating in piezoelectric layer/elastic substrate systems. Upon comparing with the predictions obtained by employing the traditional shear-lag model, the present three-layer structure model is found to be more accurate as it avoids the unrealistic displacement discontinuity across imperfectly bonded interfaces assumed by the shearlag model, especially for long waves when the piezoelectric layer is relatively thin.     

Slip pulse along an interface between two anisotropic elastic half-spaces in sliding contact with separation

The Stroh sextic formalism, together with Fourier analysis and the singular integral equation technique, is used to study the propagation of a possible slip pulse in the presence of local separation at the interface between two contact anisotropic solids. The existence of such a pulse is discussed in detail. It is found that the pulse may exist if at least one medium admits Rayleigh wave below the minimum limiting speed of the two media. The pulse-propagating speed is not fixed; it can be of any value in some regions between the lower Rayleigh wave speed and minimum limiting speed. These speed regions depend on the existence of the first and second slip-wave solutions without interfacial separation studied by Barnett, Gavazza and Lothe (Proc. R. Soc. Lond. 1988, A415, 389–419). The pulse has no free amplitude directly but involves the arbitrary size of the separation zone that depends on the intensity of the motion. The interface normal traction and the particle velocities involve a square-root singularity at both ends of the separation zones that act as energy source and sink.     

Surface electro-elastic shear horizontal waves in a layered structure with a piezoelectric substrate and a hard dielectric layer

The existence and behaviour of surface electro-elastic shear horizontal waves in a layered structure consisting of a piezoelectric substrate of crystal class 6, 4, 6mm, or 4mm mechanically bonded at its upper surface to an elastic dielectric layer and bounded by an adjacent dielectric medium is considered when the shear bulk wave velocity in the elastic layer is greater than or equal to that in the substrate. The dispersion equation for the existence of the surface electro-elastic SH waves with respect to the phase velocity is obtained which includes all the above crystal classes i.e. the surface wave problems related to all these classes are presented in a single mathematical model. The investigation of the solutions of the dispersion equation is carried out and all the possible cases of the behaviour of the surface electro-elastic SH wave depending on the electro-mechanical coefficients of the layered structure are revealed.     

Non-linear waves in a viscous fluid contained in an elastic tube with variable cross-section

In the present work, treating the large arteries as a thin-walled, long and circularly cylindrical, prestressed elastic tube with variable cross-section and using the reductive perturbation method, we have studied the amplitude modulation of non-linear waves in such a fluid-filled elastic tube. By considering the blood as an incompressible viscous fluid, the evolution equation is obtained as the dissipative non-linear Schrödinger equation with variable coefficients. It is shown that this type of equations admit a solitary wave solution with a variable wave speed. It is observed that, the wave speed increases with distance for narrowing tubes while it decreases for expanding tubes.     

Gyroscopic waves in a rotating liquid layer

The dispersion characteristics of gyroscopic waves in an incompressible liquid layer in a cavity of a rapidly rotating cylinder are studied. It is shown that in a viscous incompressible liquid layer, an inertial wave can be represented as the sum of six helical harmonics. The effects of the liquid viscosity and the ratio of the wave frequency to the angular velocity of rotation of the cylinder on the real and imaginary parts of the wavenumber are studied. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 15–20, March–April, 2008.     

Propagation of harmonic waves in longitudinally reinforced cylindrical shells with low shear stiffness

The exact solutions of the equations of motion derived under refined theories of shells and ribs based on the Timoshenko model are used to plot dispersion curves for harmonic waves propagating along a cylindrical shell reinforced with discrete longitudinal ribs __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 37–41, May 2006.     

Particle trajectories under interactions between solitary waves and a linear shear current

This paper is concerned with particle trajectories beneath solitary waves when a linear shear current exists. The fluid is assumed to be incompressible and inviscid, lying on a flat bed. Classical asymptotic expansion is used to obtain a Korteweg-de Vries(Kd V) equation, then a forth-order Runge-Kutta method is applied to get the approximate particle trajectories. On the other hand, our particular attention is paid to the direct numerical simulation(DNS) to the original Euler equations. A conformal map is used to solve the nonlinear boundary value problem. Highaccuracy numerical solutions are then obtained through the fast Fourier transform(FFT) and compared with the asymptotic solutions, which shows a good agreement when wave amplitude is small. Further, it also yields that there are different types of particle trajectories. Most surprisingly,periodic motion of particles could exist under solitary waves, which is due to the wave-current interaction.