Discontinuous wave fronts propagation in anisotropic layered media

The problem of dynamic interaction of wave phase fronts with anisotropic elastic media interfaces is considered. A technique based on joint use of the ray theory, locally plane approach and theory of stereomechanical impact is elaborated. It is employed for the investigation of discontinuous waves propagation in anisotropic tectonic structures. The cases of interaction of quasi-longitudinal and quasi-shear discontinuous waves with the interfaces separating different anisotropic elastic media are treated. The issues are considered which are associated with the wave front surfaces bifurcations, generation of their singularities and caustics, as well as with stress concentration and formation of zones where the stresses tend to infinity.     

The far-field scattering response of a side drilled hole in single/layered anisotropic media in ultrasonic pulse-echo setup

Ultrasonic testing for defects in anisotropic material is one of the difficult problems in non-destructive testing (NDT) with elastic waves. Even though a series of studies have been performed on the waves interacting with flaws, however, only a small portion was on anisotropic materials. In this paper, an analytical solution for the far-field scattering response of a side drilled hole (SDH) in anisotropic media in an ultrasonic pulse-echo setup with its incident wave normal to the axis of cylindrical hole is presented. The solution is based on the Kirchhoff approximation, and validated upon several numerical examples, yielding satisfactory results in the comparison to the results achieved by different methods. Also, an attempt is made to extend the use of this solution from homogeneous anisotropic media to weld, which is considered as the multi-layer anisotropic media.     

An eigen theory of waves in piezoelectric solids

Based on the standard spaces of the physical presentation, both the quasi-static mechanical approximation and the quasi-static electromagnetic approximation of piezoelectric solids are studied here. The complete set of uncoupled elastic wave and electromagnetic wave equations are deduced. The results show that the number and propagation speed of elastic waves and electromagnetic waves in anisotropic piezoelectric solids are determined by both the subspaces of electromagnetically anisotropic media and ones of mechanically anisotropic media. Based on these laws, we discuss the propagation behaviour of elastic waves and electromagnetic waves in the piezoelectric material of class 6 mm.     

Reflection and refraction of plane quasi-longitudinal waves at an interface of two piezoelectric media under initial stresses

This paper is devoted to study a problem of reflection and refraction of quasi-longitudinal waves under initial stresses at an interface of two anisotropic piezoelectric media with different properties. One of the two media is aluminum nitride, which is considered the down piezoelectric medium and the above medium is chosen as PZT-5H ceramics. The two piezoelectric media welded are assumed to be anisotropic of a type of a transversely isotropic crystals (hexagonal crystal structure, class 6 mm). The equations of motion and constitutive relations for the piezoelectric media have been written. Suitable boundary conditions are used to obtain the reflection and refraction coefficients. For an incidence of quasi-longitudinal plane waves, four independent-type amplitude ratios of elastic displacement components for plane waves, called quasi-longitudinal (qP) and quasi-shear vertical (qSV) waves, are shown to exist. Also, it is observed that there exist four dependent amplitude ratios of electric potential, which are proportional to the previous four types. Finally, it is found that the coefficients of reflection and refraction are functions of angle of incidence, elastic constants, piezoelectric potential parameters and the initial stresses. Numerical computations and the results obtained are depicted graphically. In the end, a particular case has been reduced from the present study. This investigation is considered important because the initial stresses in such practical problems are inevitable and may result in frequency shift, a change in the velocity of surface waves and controlling the selectivity of a filter compensation of the devices.     

Kinetic modeling of multiple scattering of elastic waves in heterogeneous anisotropic media

In this paper we develop a multiple scattering model for elastic waves in random anisotropic media. It relies on a kinetic approach of wave propagation phenomena pertaining to the situation whereby the wavelength is comparable to the correlation length of the weak random inhomogeneities—the so-called weak coupling limit. The waves are described in terms of their associated energy densities in the phase space position  ×$\times$  wave vector. They satisfy radiative transfer equations in this scaling, characterized by collision operators depending on the correlation structure of the heterogeneities. The derivation is based on a multi-scale asymptotic analysis using spatio-temporal Wigner transforms and their interpretation in terms of semiclassical operators, along the same lines as Bal (2005). The model accounts for all possible polarizations of waves in anisotropic elastic media and their interactions, as well as for the degeneracy directions of propagation when two phase speeds possibly coincide. Thus it embodies isotropic elasticity which was considered in several previous publications. Some particular anisotropic cases of engineering interest are derived in detail.     

Reflection and refraction of plane waves on the interface of two anisotropic media

We use generalized functionally invariant solutions [1] of the equations of motion to obtain and analytically study solutions of the plane problem of reflection and refraction of plane waves on the interface of two anisotropic media with four elastic constants depending on the angles of incidence of primary waves for various relations between the elastic constants of the contacting media. For the primary waves, we find the ranges of incidence angles for which real and complex secondary waves are excited. We study all possible combinations of the distribution of phase velocities and reflection and refraction angles in detail and obtain conditions characterizing the directions of the energy flux vectors for the primary and secondary waves depending on the incidence angles of the primary waves for different relations between the elastic constants of the contacting media, which satisfy the necessary and sufficient conditions for the elastic energy form to be positive definite.     

Elastic waves in crystals and anisotropic media

The results of investigation of propagation of elastic waves in anisotropic media are discussed taking into account the two-dimensional problem of a source in an infinite medium and the Lamb problem for a half-plane. The media considered in the investigation are those for which the equations of motion under plane deformation conditions are characterized by four constants.     

On the investigation of elasticity equation for orthotropic materials and the solution of the associated scattering problem

The present work concerns the investigation of the two-dimensional direct scattering problem of time-harmonic elastic waves from bounded anisotropic components of isotropic media. We obtain a Fourier series expansion for the elastic field in the interior of the anisotropic inclusion based on a suitable diagonalization applied to the underlying differential system and a plane wave expansion of the sought field, provided that the inclusion exhibits orthotropic symmetry. This expansion is then exploited to acquire a semi-analytical solution to the associated elastic transmission scattering problem. Numerical results for several geometric configurations and varying degree of anisotropy are presented revealing the pronounced effect of the specific anisotropic character on the scattering mechanism.     

Criteria for elastic waves in anisotropic media — a consolidation

The geometry relating to the tangent plane at a stationary point on a surface has been used to re-examine various criteria for the existence of parabolic points (inflexions in 2D-sections) on the outermost sheet of the slowness surface for elastic waves in anisotropic media.Previous results obtained by the authors, exact [4,5], and approximate [6], are related in detail. Further approximations, based on geometrical properties, are derived; one of these proves equivalent to a sufficient condition first applied by McCurdy [8].Numerical investigation shows that, over a wide range of anisotropy, the simply applied approximate criterion of [6] is sufficient and within the accuracy of observation of the elastic stiffnesses.     

Elastodynamic analysis of inhomogeneous anisotropic bodies

The integral equation method is presented for elastodynamic problems of inhomogeneous anisotropic bodies. Since fundamental solutions are not available for general inhomogeneous anisotropic media, we employ the fundamental solution for homogeneous elastostatics. The terms induced by material inhomogeneity and inertia force are regarded as body forces in elastostatics, and evaluated in the form of volume integrals. The scattering problems of elastic waves by inhomogeneous anisotropic inclusions are investigated for some test cases. Numerical results show the significant effects of inhomogeneity and anisotropy of materials on wave propagations.     

Partial Uniqueness and Obstruction to Uniqueness in Inverse Problems for Anisotropic Elastic Media

We consider the inverse problem of identifying the density and elastic moduli for three-dimensional anisotropic elastic bodies, given displacement and traction measurements made at their surface. These surface measurements are modelled by the dynamic Dirichlet-to-Neumann map on a finite time interval. For linear or nonlinear anisotropic hyperelastic bodies we show that the displacement-to-traction surface measurements do not change when the density and elasticity tensor in the interior are transformed tensorially by a change of coordinates fixing the surface of the body to first order. Our main tool, a new approach in inverse problems for elastic media, is the representation of the equations of motion in a covariant form (following Marsden and Hughes, 1983) that preserves the underlying physics.In the case of classical linear elastodynamics we then investigate how the type of anisotropy changes under coordinate transformations. That is, we analyze the orbits of general linear, anisotropic elasticity tensors under the action by pull-back of diffeomorphisms that fix the surface of the elastic body to first order, and derive a pointwise characterization of parts of the orbits under this action. For example, we show that the orbit of isotropic elastic media, at any point in the body, consists of some transversely isotropic and some orthotropic elastic media. We then derive the first uniqueness result in the inverse problem for anisotropic media using surface displacement-traction data: uniqueness of three elastic moduli for tensors in the orbit of isotropic elasticity tensors. ^†Partially supported by an MSRI Postdoctoral Fellowship. Research at MSRI is supported in part by NSF grant DMS-9850361. This work was conducted while the first author was a Gibbs Instructor at Yale University. ^‡Partially supported by an MSRI Postdoctoral Fellowship, and by NSF grant DMS-9801664 (9996350).     

The mathematical model of reflection and refraction of longitudinal waves in thermo-piezoelectric materials

In this paper, the basic equations of motion, of Gauss and of heat conduction, together with constitutive relations for pyro- and piezoelectric media, are presented. Three thermoelastic theories are considered: classical dynamical coupled theory, the Lord–Shulman theory with one relaxation time and Green and Lindsay theory with two relaxation times. For incident elastic longitudinal, potential electric and thermal waves, referred to as qP, φ-mode and T-mode waves, which impinge upon the interface between two different transversal isotropic media, reflection and refraction coefficients are obtained by solving a set of linear algebraic equations. A case study is investigated: a system formed by two semi-infinite, hexagonal symmetric, pyroelectric–piezoelectric media, namely Cadmium Selenide (CdSe) and Barium Titanate (BaTiO₃). Numerical results for the reflection and refraction coefficients are obtained, and their behavior versus the incidence angle is analyzed. The interaction with the interface give rises to different kinds of reflected and refracted waves: (i) two reflected elastic waves in the first medium, one longitudinal (qP-wave) and the other transversal (qSV-wave), and a similar situation for the refracted waves in the second medium; (ii) two reflected potential electric waves and a similar situation for the refracted waves; (iii) two reflected thermal waves and a similar situation for the refracted waves. The amplitudes of the reflected and refracted waves are functions of the incident angle, of the thermal relaxation times and of the media elastic, electric, thermal constants. This study is relevant to signal processing, sound systems, wireless communications, surface acoustic wave devices and military defense equipment.     

Transient Waves on the Free Surface of a Viscoelastic Anisotropic Half-Space

The Stroh formalism is employed to discuss the existence of transient surface waves on a viscoelastic anisotropic hall-space. The compatibility conditions, obtained using the integral formulation of Lothe and Barnett [13, 14], are examined on the basis of an asymptotic expansion of the viscoelastic kernel and a separation of space variables. Some previous results on elastic media are extended to viscoelasticity, exploiting the consequences of the second law of thermodynamics. It is found that all the allowed transient surface modes take the form of inhomogeneous plane waves whose amplitude exponentially decays along the propagation direction on the surface. Special solutions are derived explicitly for one-component surface waves where transient modes are admitted also in those cases in which stationary waves cannot occur. Mathematics Subject Classifications (2000) 74D05, 74J15.     

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.     

An analysis of thermally-induced plane waves in elastic-plastic single crystals

A framework for the calculation of thermally-induced plane waves in elastic-plastic single crystals of arbitrary crystallographic symmetry and orientation is presented. Plasticity is described in terms of small strain theory and the available slip-planes which can be arbitrary in number as well as in orientation. The effects of perfect-plasticity modify not only the anisotropic elastic moduli, but also the components of the Grüneisen tensor. The latter effect is a consequence of a non-spherical stress state developed in anisotropic materials during rapid energy-absorption at constant strain. Specific examples of thermally-induced plane waves are presented for both the elastic and plastic response of beryllium and graphite single-crystals.     

On the Existence of Longitudinal Plane Waves in General Elastic Anisotropic Media

We give a new proof of Kolodner's result that longitudinal waves can propagate in at least three directions in a hyperelastic anisotropic medium. We give examples of an orthotropic hyperelastic tensor with exactly three such directions, of a monoclinic elastic (but not hyperelastic) tensor with only one, and of a monoclinic elastic (elliptic, but not uniformly elliptic) tensor with no direction for longitudinal waves. This revised version was published online in August 2006 with corrections to the Cover Date.     

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.     

Seismic Wave Propagation in Composite Elastic Media

It has been known since the time of Biot–Gassman theory (Biot, J Acoust Soc Am 28:168–178, 1956, Gassmann, Naturf Ges Zurich 96:1–24, 1951) that additional seismic waves are predicted by a multicomponent theory. It is shown in this article that if the second or third phase is also an elastic medium then multiple p and s waves are predicted. Futhermore, since viscous dissipation no longer appears as an attenuation mechanism and the media are perfectly elastic, these waves propagate without attenuation. As well, these additional elastic waves contain information about the coupling of the elastic solids at the pore scale. Attempts to model such a medium as a single elastic solid causes this additional information to be misinterpreted. In the limit as the shear modulus of one of the solids tends to zero, it is shown that the equations of motion become identical to the equations of motion for a fluid filled porous medium when the viscosity of the fluid becomes zero. In this limit, an additional dilatational wave is predicted, which moves the fluid though the porous matrix much similar to a heart pumping blood through a body. This allows for a connection with studies which have been done on fluid-filled porous media (Spanos, 2002).     

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.     

On shapes of body waves in periodically inhomogeneous, magnetostrictive, dielectric materials

The shapes of shear body waves in periodically inhomogeneous, magnetostrictive, dielectric media are studied with emphasis on the partial (elastic and magnetostrictive) wave motions coupled to produce magnetoelastic waves __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 7, pp. 57–63, July 2006.