Surface and interfacial waves in anisotropic elastic quasicrystals

We present the Stroh formalism for two-dimensional subsonic steady-state motion of anisotropic quasicrystals. Using this new formalism and a series of identities and properties which follow, we investigate subsonic surface and interfacial waves in anisotropic quasicrystals. Our results suggest that there exist at most three subsonic surface wave speeds. This interesting observation is quite different from the unique surface wave speed known for anisotropic crystals. The degenerate case of decagonal quasicrystalline materials is discussed in detail.     

Explicit conditions for the existence of exceptional body waves and subsonic surface waves in anisotropic elastic solids

It is known that a subsonic surface (Rayleigh) wave exists in an anisotropic elastic half-space x₂  0 if the first transonic state is not of Type 1. If the first transonic state is of Type 1 but the limiting wave is not exceptional, a subsonic surface wave exists. If the first transonic state is of Type 1 and the limiting wave is exceptional, a subsonic surface wave exists when . It is shown that an exceptional body wave is necessarily an exceptional transonic wave, and could be an exceptional limiting wave. Only two wave speeds are possible for an exceptional body wave. We present explicit conditions in terms of the reduced elastic compliances for the existence of an exceptional body wave. If an exceptional body wave exists, conditions are given for identifying whether the transonic state is of Type 1. Hence, through the existence of an exceptional body wave we provide explicit conditions for the existence of a subsonic surface wave with the exception when needs to be computed.     

Purely transverse waves in elastic anisotropic media

Formulas are obtained for decompositions of the third- and fourth-rank tensors symmetric in the last two and three indices, respectively, into irreducible parts invariant relative to the orthogonal group of coordinate transformation. The corresponding parts of the decompositions are orthogonal to each other. These decompositions are used to obtain a general representation of the displacement vectors of plane transverse waves in elastic isotropic and anisotropic solids. It is shown that the displacement vectors of transverse waves are second-, third-, and fourth-degree homogeneous polynomials of the wave normal. Special orthotropic materials are found that transmit purely transverse waves for any direction of the wave normal. The eigenmoduli, eigenstates, and engineering constants (bulk moduli, Youngs moduli, Poissons ratios, shear moduli, and Lame constants of the closest isotropic materials) are determined for these materials.Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 46, No. 1, pp. 160–172, January–February, 2005     

Nonlinear guided bulk waves in heterogeneous elastic structural elements

Guided nonlinear bulk strain waves are considered in heterogeneous composite materials, containing macro- or micro(nano)inclusions as the main sources of nonuniformity. Theory is based on the combination of the finite deformation and the 5-constant (Murnaghan) nonlinear elasticity, that leads under several assumptions to the only nonlinear doubly dispersive equation with variable coefficients (DDE) for a component of longitudinal strain in rod and in shell. Numerical simulations performed and experimental data are discussed, and lead to conclusions concerning dramatic influence of heterogeneity in dynamic behaviour of strain solitons in solids. We have shown how different smooth variations in elasticity may lead to either amplification or decay of strain solitons in both polymer rod and shell. These date can be useful in the NDT problems, and in problems of the solids integrity under intense elastic pulse loading.     

Implicit and explicit secular equations for Rayleigh waves in two-dimensional anisotropic media

This paper is concerned with the derivation of implicit and explicit secular equations for Rayleigh waves polarized in a plane of symmetry of an anisotropic linear elastic medium. It has been confirmed, in accord with Ting’s paper [2], that the Rayleigh waves propagate with no geometric dispersion. Numerical evaluations of both the implicit and explicit equations give the same values of Rayleigh wave velocities. In the case of orthotropic material (thin composites) it has been found that Rayleigh wave velocity depends significantly (as with bulk waves) on the directions of principal material axes. For the same material model the analytical solutions, based on implicit and explicit secular equations, were compared against the finite element and experimental data that had been published by Cerv et al. [4] in 2010. It emerged that the theory was in accordance with the experiment.     

Multiple scattering of elastic waves by pinned dislocation segments in a continuum

The coherent propagation of elastic waves in a solid filled with a random distribution of pinned dislocation segments is studied to all orders in perturbation theory. It is shown that, within the independent scattering approximation, the perturbation series that generates the mass operator is a geometric series that can thus be formally summed. A divergent quantity is shown to be renormalizable to zero at low frequencies. At higher frequencies said quantity can be expressed in terms of a cut-off with dimensions of length, related to the dislocation length, and physical quantities can be computed in terms of two parameters, to be determined by experiment. The approach used in this problem is compared and contrasted with the scattering of de Broglie waves by delta-function potentials as described by the Schrödinger equation.     

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$x_2\le 0$.     

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.     

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p² with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x₂ of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.     

Continuum Dyson's equation and defect Green's function in a heterogeneous anisotropic solid

A continuum Dyson's equation and a defect Green's function (GF) in a heterogeneous, anisotropic and linearly elastic solid under homogeneous boundary conditions have been introduced. The continuum Dyson's equation relates the point-force Green's responses of two systems of identical geometry and boundary conditions but of different media. Given the GF of either system (i.e., a reference), the GF of the other (i.e., a defect system with “defect” change of materials property relative to the reference) can be obtained by solving the Dyson's equation. The defect GF is applied to solve the eigenstrain problem of a heterogeneous solid. In particular, the problem of slightly inhomogeneous inclusions is examined in detail. Based on the Dyson's equation, approximate schemes are proposed to efficiently evaluate the elastic field. Numerical results are reported for inhomogeneous inclusions in a semi-infinite substrate with a traction-free surface to demonstrate the validity of the present formulation.     

Acoustic axes in elasticity

New results are presented for the degeneracy condition of elastic waves in anisotropic materials. The condition for the existence of acoustic axes involves a traceless symmetric third order tensor that must vanish identically. It is shown that all previous representations of the degeneracy condition follow from this acoustic axis tensor. The conditions for existence of acoustic axes in elastic crystals of orthorhombic, tetragonal, hexagonal and cubic (RTHC) symmetry are reinterpreted using the geometrical methods developed here. Application to weakly anisotropic solids is discussed, and it is shown that the satisfaction of the acoustic axes conditions to first order in anisotropy does not in general coincide with true acoustic axes.     

弹性波散射与多重散射的T矩阵方法

研究弹性波散射与多重散射的T矩阵方法。首先,基于Helmholtz体内和体外公式推导了对应于圆柱型散射体的T矩阵元素的具体表达式;接着分析了在含多个随机分布圆柱型散射体的随机非均匀介质中弹性波的多重散射并给出在统计平均意义下的相干波的定义以及波速和衰减系数计算公式;最后,针对Ge/Al、Sic/Al复合材料用Matlab进行了编程和数值计算;计算单个柱型散射体的散射截面以及随机非均匀介质中相干波的速度和衰减系数,分析了这种介质的频散特性。     

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.     

Rayleigh waves of arbitrary profile in anisotropic media

The paper deals with surface wave propagation in an orthorhombic elastic half-plane. The general profile of the wave is considered, incorporating the anisotropy effects within the known representation in terms of a single plane harmonic function.     

Rayleigh waves in anisotropic porous media and the polarization vector method

In this paper, the propagation of Rayleigh waves in orthotropic non-viscous fluid-saturated porous half-spaces with sealed surface-pores and with impervious surface is investigated. The main aim of the investigation is to derive explicit secular equations and based on them to examine the effect of the material parameters and the boundary conditions on the propagation of Rayleigh waves. By employing the method of polarization vector the explicit secular equations have been derived. These equations recover the ones corresponding to Rayleigh waves propagating in purely elastic half-spaces. It is shown from numerical examples that the Rayleigh wave velocity depends strongly on the porosity, the elastic constants, the anisotropy, the boundary conditions and it differs considerably from the one corresponding to purely elastic half-spaces. Remarkably, in the fluid saturated porous half-spaces, Rayleigh waves may travel with a larger velocity than that of the shear wave, a fact that is impossible for the purely elastic half-spaces.     

Scattering attenuation of elastic waves due to low-contrast inclusions

Seismic scattering attenuation due to random lithospheric heterogeneity has been theoretically modeled using two approaches. One approach is the Born approximation theory (BAT), which is primarily used to treat weak continuous heterogeneity, and the other approach is the Foldy approximation theory (FAT), which deals with sparsely distributed discrete inclusions. We apply the BAT to elastic wave scattering due to inclusions having low contrast with the matrix, and compare the results with those predicted by the FAT. We thus investigate the valid wavenumber range of the BAT based on a reasonable assumption that the inclusions are distributed so sparsely that the FAT is effectively correct for any wavenumber. For simplicity, we consider a specific type of round inclusion, which is either two- or three-dimensional and has a two-valued wave velocity and/or mass density. Both theories are confirmed to yield essentially equivalent results below a certain wavenumber limit, depending on the contrast. This is known as the Rayleigh-Gans scattering regime. Beyond the wavenumber limit, the BAT overestimates the attenuation for common-mode scattering due to wave-velocity contrast, but remains valid with respect to the attenuation for scattering due to mass-density contrast and/or conversion scattering. These conclusions are independent of the spatial dimensions of the media as well as the modes of the elastic waves (P or S). Some advantages of the BAT over the FAT for application to low-contrast inclusions are discussed.     

The anisotropic and angularly inhomogeneous elastic wedge under a monomial load distribution

In this study, the generally anisotropic and angularly inhomogeneous wedge under a monomial type of distributed loading of order n of, the radial coordinate r at its external faces is considered. At first, using variable separable relations in the equilibrium equations, the strain–stress relations and the strain compatibility equation, a differential system of equations, is constructed. Decoupling this system, an ordinary differential equation is derived and the stress and displacement fields may be determined. The proposed procedure is also applied to the elastostatic problem of an isotropic and angularly inhomogeneous wedge. The special cases of loading of order n=−1 and n=−2, where the self-similarity approach is not valid, are examined and the stress and displacements fields are derived. Applications are presented for the cases of an angularly inhomogeneous wedge and in the case of a bi-material isotropic wedge.     

基于积分方程方法的弹性波多重散射计算

运用积分方程方法计算了含多个随机分布椭圆柱型孔洞的随机非均匀介质中相干波的速度和衰减系数,分析了这种介质的频散特性。首先,建立了散射位移场满足的积分方程,推导了单个椭圆柱孔洞的散射截面计算公式。接着分析了在含多个随机分布椭圆柱型孔洞的随机非均匀介质中弹性波的多重散射,给出在统计平均意义下的相干波的波速和衰减系数计算公式。然后用Matlab进行了编程,给出了一个数值算例,并将计算结果与波函数展开法进行了比较,分析了随机空隙介质的频散特征及其孔洞椭圆偏心率和材料空隙率的影响。